Integration 1_________________________________________________________________________________

myarctan := proc(y,x) `if`(nargs = 1, arctan(y),evalf(Pi/2)-arctan(expand(x/y))); end: myarctanh := (y) -> myln(1+y)/2 - myln(1-y)/2: myln := proc(u) local c,d,v; d := degree(u,x); c := coeff(u,x,d); v := expand(x^d + (u-c*x^d)/c); if d = 1 then v := abs(v); fi; ln(v); end:ratint := proc(f,x) local g; g := int(f,x); if has(g,{RootOf,int}) then g := int(convert(evalf(f),parfrac,x),x); fi; g := eval( subs(ln=myln,arctan = myarctan,arctanh=myarctanh, signum=1,abs=(x->x), g)); g := expand(g); if (type(g,`+`)) then g := remove((a)->type(a,constant),g); fi; g; end:

g[1] := (x) -> x^2 + x + 1 + 1/x + 1/x^2;g[2] := (x) -> 1/(x-1) + 1/(x^2-1) + 1/(x^3-1);g[3] := (x) -> 1/(x-1)^2 + 1/(x-2)^3 + 1/(x-3)^4;g[4] := (x) -> 1/(x+1) + 1/(x^2+1) + 1/(x^3+1);g[5] := (x) -> x/(x+1)*(x+2)/(x+3)*(x+4);g[6] := (x) -> x/(x+1)^2*(x+2)/(x+3)^2*(x+4);g[7] := (x) -> (x^3+4*x^2+x+2)/(x-1)^2/(x+1)^3;g[8] := (x) -> 1/(x^2+8*x+1);g[9] := (x) -> (x^2+1)^(-5);g[10] := (x) -> 1/(x^8+x^2+1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIiZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCwqJDkkIiIjRjBGOUYwRjBGMCokRjkhIiJGMCokRjkhIiNGMEYoRihGKDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIjZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCgqJCwmOSQiIiIhIiJGO0Y8RjsqJCwmKiRGOkYwRjtGPEY7RjxGOyokLCYqJEY6IiIkRjtGPEY7RjxGO0YoRihGKDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIkZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCgqJCwmOSQiIiIhIiJGOyEiI0Y7KiQsJkY6RjtGPUY7ISIkRjsqJCwmRjpGO0ZARjshIiVGO0YoRihGKDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIlZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCgqJCwmOSQiIiJGO0Y7ISIiRjsqJCwmKiRGOiIiI0Y7RjtGO0Y8RjsqJCwmKiRGOiIiJEY7RjtGO0Y8RjtGKEYoRig3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiImZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiw5JCIiIiwmRjhGOUY5RjkhIiIsJkY4RjkiIiNGOUY5LCZGOEY5IiIkRjlGOywmRjhGOSIiJUY5RjlGKEYoRig3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiInZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiw5JCIiIiwmRjhGOUY5RjkhIiMsJkY4RjkiIiNGOUY5LCZGOEY5IiIkRjlGOywmRjhGOSIiJUY5RjlGKEYoRig3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKigsKiokOSQiIiQiIiIqJEY6IiIjIiIlRjpGPEY+RjxGPCwmRjpGPCEiIkY8ISIjLCZGOkY8RjxGPCEiJEYoRihGKDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIpZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQsKCokOSQiIiMiIiJGOkYwRjxGPCEiIkYoRihGKDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiIqZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQsJiokOSQiIiMiIiJGPEY8ISImRihGKEYoNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiZ0dGKDYjIiM1Zio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQsKCokOSQiIikiIiIqJEY6IiIjRjxGPEY8ISIiRihGKEYoNyNGMQ==

______________________________________________________________________________Exercise 1When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ==. Indeed, the formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is as follows: y := ratint(((x^3+1)/(x^2+1))^3,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLDQqJEkieEdGKCIiJSMiIiJGMSokRjAiIiMjISIkRjVGMCIiJComLCZGNEYzRjNGMyEiI0YwRjgjIiM6RjEqJkY6RjtGMEY1I0YzRjUqJkY6RjtGMEYzIyIjOEYxKiRGOkY7RjMtSSNsbkdGJTYjRjpGOC1JJ2FyY3RhbkdGJTYjRjAjISNARjE3I0Yu

The first term is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ==, and the other terms are much smaller, at least when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large. Indeed, when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large, all of the terms with 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 on the bottom will be very small. Also, you should recall from the theory of trigonometric functions that 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 is always at most LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjctRiM2JC1JI21uR0YkNiRRIjJGJ0Y3RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkUvJSliZXZlbGxlZEdGNkY3, so this will also be negligible in comparison to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== for large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYkUSIzRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjsvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJ4RidGL0YyRjtGK0Y7 and 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 will get large as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= does, but much more slowly than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== does. For example, when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRJTEwMDBGJ0Y+Rj5GK0Y+, the term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== is of the order of a trillion, whereas 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 is of the order of a million, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYkUSIzRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjsvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJ4RidGL0YyRjtGK0Y7 is only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMzAwMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. The behaviour of 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 is a little less obvious, but in fact it grows much more slowly even than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, as you could check by plotting the graph. We can ask Maple to separate the terms as follows: terms := [op(y)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0ZXJtc0dGKDcrLCQqJEkieEdGKCIiJSMiIiJGMiwkKiRGMSIiIyMhIiRGNywkRjEiIiQsJComLCZGNkY0RjRGNCEiI0YxRjsjIiM6RjIsJComRj5GP0YxRjcjRjRGNywkKiZGPkY/RjFGNCMiIzhGMiokRj5GPywkLUkjbG5HRiU2I0Y+RjssJC1JJ2FyY3RhbkdGJTYjRjEjISNARjI3I0Yu

We can then check the relative size when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRJTEwMDBGJ0Y+Rj5GK0Y+: evalf(subs(x=1000,terms));

NyskIisrKysrRCIiIyQhKCsrXSIiIiEkIiUrSUYoJCIrK0QqKlxQISM3JCIrKysqKioqXCEjOyQiKytOKipcSyEjPSQiKysrKSoqKioqISNBJCIrb01sV1QhIikkISs8MlZUIykhIio=

You can see that the first term is much bigger than the rest. We can also see this by plotting. In the picture below, the red curve is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ==, and the other terms are plotted in various different colours.By LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjVGJ0Y+Rj5GK0Y+, the red curve is already much bigger than the others. If you change the range to 1..100 (say) then only the red curve is visible because all other terms are squashed down onto the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis. plot(terms,x=1..5);

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We can also just plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjRGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JEY9RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== together. If we stop at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjlGJ0Y+Rj5GK0Y+ then you can still see the difference between the two curves, but if we went much further than that then they would be indistinguishable. plot([y,x^4/4],x=-9..9);

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unassign('y','terms'); _________________________________________________________________________________Exercise 2Here is the formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: y := ratint((a*x^2+b)/(c*x^2+d),x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqKEkiYUdGKCIiIkkiY0dGKCEiIkkieEdGKEYxRjEqLEYyRjMqJkkiZEdGKEYxRjJGMSNGMyIiIy1JJ2FyY3RhbkdGJTYjKihGMkYxRjRGMUY2RjhGMUYwRjFGN0YxRjMqKEY2RjhGOkYxSSJiR0YoRjFGMTcjRi4=

This is 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 (which gives a line of slope LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRictRiM2JC1GMTYlUSJjRidGNEY3RjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkYvJSliZXZlbGxlZEdRJmZhbHNlRidGOg==) plus some other terms, which will be much smaller when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large. One way to see this is to look at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnLUYjNiQtRjE2JVEjZHhGJ0Y0RjdGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is large, or in other words, the limit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnLUYjNiQtRjE2JVEjZHhGJ0Y0RjdGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tends to infinity. Of course LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnLUYjNiQtRjE2JVEjZHhGJ0Y0RjdGOi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6 is just the function we first thought of, namely 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. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is very large, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= will be negligible compared with 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 and 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, so the function is approximately 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. limit(diff(y,x),x=infinity);

KiZJImFHNiIiIiJJImNHRiQhIiI=

We can see this graphically if we choose some numbers for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY1LUYsNiVRImNGJ0YvRjJGOQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. plot(subs(a=7,b=-3,c=2,d=8,[y,a*x/c]),x=10..100);

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unassign('y'); _________________________________________________________________________________Exercise 3(a) It is true that the integral of a rational function can contain terms like 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. For example: y := (8*x+8)/(x^2+2*x+3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoKiYsJkkieEdGKCIiKUYxIiIiRjIsKCokRjAiIiNGMkYwRjUiIiRGMiEiIjcjRi4=

ratint(y,x);

LCQtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCgqJEkieEdGKCIiIyIiIkYsRi0iIiRGLiIiJQ==

or more generally: y := a*(2*x+u)/(x^2+u*x+v);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoKihJImFHRigiIiIsJkkieEdGKCIiI0kidUdGKEYwRjAsKComRjRGMEYyRjBGMCokRjJGM0YwSSJ2R0YoRjAhIiI3I0Yu

ratint(y,x);

KiZJImFHNiIiIiItSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJkkidUdGJEYlSSJ4R0YkRiVGJSokRi8iIiNGJUkidkdGJEYlRiU=

(b) It is not true that terms like 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 can occur in the integral of a rational function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4=. If there were a term 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 in 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, then there would have to be a term 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 in 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. In other words, there would be a term 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 in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4=, which is not allowed, as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4= is supposed to be a rational function.(c) For essentially the same reason, there cannot be any terms like 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 in the integral of a rational function, because 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, which is not allowed asa term in a rational function._________________________________________________________________________________Exercise 4If 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 then the function 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 has no real roots, and the integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiktSSNtb0dGJDYtUSgmIzg3NDc7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIxRidGO0Y7LUYjNiZGKy1GIzYpRistRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUZXNiRRIjJGJ0Y7LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y7LUY4Ni1RIitGJ0Y7Rj1GQEZCRkRGRkZIRkovRk1RLDAuMjIyMjIyMmVtRicvRlBGam8tRiM2Ji1GLDYlUSJiRidGL0YyLUY4Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y7Rj1GQEZCRkRGRkZIRkpGTEZPRl1vRjtGZm8tRiw2JVEiY0YnRi9GMkY7RitGOy8lLmxpbmV0aGlja25lc3NHRlkvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW3EvJSliZXZlbGxlZEdGP0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRmVxLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY4Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjtGPUZARkJGREZGRkhGSkZMRk9GXW9GO0YrRjs= involvesthe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEnYXJjdGFuRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGPC8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItSShtZmVuY2VkR0YkNiQtRiM2JEYrRjxGPEY8RitGPA== function. Here is an example: b := 2; c := 3; b^2-4*c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJiR0YoIiIjNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoIiIkNyNGLg==

ISIp

plot(x^2+b*x+c,x=-2..1,0..6);

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ratint(1/(x^2+b*x+c),x);

LCQqJiIiIyMiIiJGJC1JJ2FyY3Rhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJkYkRiVJInhHRixGJkYlKiRGJEYlRiVGJkYl

If 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 then the function 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 has only one real root (where the graph touches the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis but does not cross it) and the integral involves neither LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEnYXJjdGFuRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGPC8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItSShtZmVuY2VkR0YkNiQtRiM2JEYrRjxGPEY8RitGPA== nor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEjbG5GJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JKG1mZW5jZWRHRiQ2JC1GIzYkRitGPEY8RjxGK0Y8; in fact, it justhas the form 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. Here is an example: b := 4; c := 4; b^2-4*c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJiR0YoIiIlNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoIiIlNyNGLg==

IiIh

plot(x^2+b*x+c,x=-4..1,0..6);

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ratint(1/(x^2+b*x+c),x);

LCQqJCwmSSJ4RzYiIiIiIiIjRichIiJGKQ==

If 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 then the function 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 has two real roots, and the integral is asum of two terms involving LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEjbG5GJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JKG1mZW5jZWRHRiQ2JC1GIzYkRitGPEY8RjxGK0Y8. Here is an example: b := -5; c := 4; b^2-4*c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJiR0YoISImNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoIiIlNyNGLg==

IiIq

plot(x^2+b*x+c,x=-1..5);

6*-%'CURVESG6$7dw7$$!#5!""$"$+"!""7$$!1jC\)pVXo*!#;$"1Jjs:d<!y*!#:7$$!1KkGdo15%*!#;$"1A#H7,F0f*!#:7$$!1nNrU*)Q,"*!#;$"1sAAys/z$*!#:7$$!1jFb5\m!z)!#;$"1<E(4N!4o"*!#:7$$!1(Qxap<9[)!#;$"1j-**3L0g*)!#:7$$!1<NqSfq%>)!#;$"1ux*[-&))o()!#:7$$!1T"Gc_Ky*y!#;$"1pF%)[Qns&)!#:7$$!1x`2:c!3f(!#;$"1(G'R1hgr$)!#:7$$!1$f=PMjZG(!#;$"1e"*fT%fI<)!#:7$$!1c7D]7'*pp!#;$"1pcUBUyqz!#:7$$!1u[(\R$o#p'!#;$"1i6u2=E%z(!#:7$$!1w],.U`!Q'!#;$"0*fmo#ztf(!#97$$!17C['H.r1'!#;$"2M3VB2\;S(!#;7$$!1=NqSX0ld!#;$"1AY#*3e)[@(!#:7$$!1U$oOLl2\&!#;$"0JGjqno/(!#97$$!1<MoO*3Y;&!#;$"0TIIKO!\o!#97$$!1h@V'o6$))[!#;$"1sJla<6$o'!#:7$$!2D^-0I7pc%!#<$"2v1c)HI-#\'!#;7$$!1CY#\Q>CG%!#;$"1%4Y.&3gCj!#:7$$!1rT$oc&GqR!#;$"1-#R#eWxUh!#:7$$!2.(Rze.1tO!#<$"1A/R.RWrf!#:7$$!1ze<Nq$HO$!#;$"1:Nhs>c%z&!#:7$$!2:NqS,Z"yI!#<$"1yuV(RBQj&!#:7$$!1jD^-D'4x#!#;$"1)efJejAY&!#:7$$!24>Qw7%)=X#!#<$"1PFd@%fgG&!#:7$$!14;Kka7u@!#;$"1@p]Z4LM^!#:7$$!2DY#\)\QT(=!#<$"1Z:O+K>s\!#:7$$!2rAX!4EAk:!#<$"1'4H#G#zl![!#:7$$!2uMpQPI5E"!#<$"2X)G(H;<kk%!#;7$$!1ET#['Hww'*!#<$"1X*)QA@?$\%!#:7$$!1nAX!4m&>k!#<$"1#e,uQ*4DV!#:7$$!1Qd9H)HG\$!#<$"1C)yuZhe<%!#:7$$!1xFa3<9zO!#=$"1DkiW#4%=S!#:7$$"2MsY$pQsjC!#=$"2F5j+!3UxQ!#;7$$"1BX!4=_%fb!#<$"0&*4=9=^s$!#97$$"1W;Lm7Hs%)!#<$"1uNh3Mc$e$!#:7$$"2k^.2u/<:"!#<$"2;$eun=TPM!#;7$$"2BmKlI&Q\9!#<$"2mf,W_9jH$!#;7$$"1#yc8FH5w"!#;$"1jtE0w\]J!#:7$$"1JiC\%z61#!#;$"2Y$of#)[*=,$!#;7$$"2W%*)ydj7oB!#<$"2`2qb/<?(G!#;7$$"1uZ&4fJDn#!#;$"2d7pbXe^t#!#;7$$"1['HfQ[A&H!#;$"1=uN7N.6E!#:7$$"2a(\**)RNGF$!#<$"2&e;*f"opqC!#;7$$"08E_Cs&fN!#:$"1xn2B%>pM#!#:7$$"2m@V'GNHlQ!#<$"1.TIw"en@#!#:7$$"0:Ig!y!z:%!#:$"2%*Q>-2GR4#!#;7$$"1&*)ydN=G[%!#;$"2[j>iUZ&f>!#;7$$"20X!4=q;kZ!#<$"2KQ<ZO*)[%=!#;7$$"1e;Lm-]$3&!#;$"2667khpmr"!#;7$$"1#Qw_&y`u`!#;$"2uzRRk(e,;!#;7$$"1")f>Rs)Gp&!#;$"2O,%eJgkx9!#;7$$"1jHf=*ov'f!#;$"2m3&oGVLs8!#;7$$"16AW)G*R!G'!#;$"2([:+3XBa7!#;7$$"1[&4>)zd#e'!#;$"2*HW#zL9?9"!#;7$$"1LnMp#fX)o!#;$"2R6e'G>pJ5!#;7$$"1`06A'Ha=(!#;$"0ix[x"*eB*!#:7$$"105?S=Zuu!#;$"1`2*zs8W@)!#;7$$"1<LmKT$py(!#;$"19rKbO'*Gr!#;7$$"1X)oPNLZ3)!#;$"1:+JrXi7h!#;7$$"1w_06I:)R)!#;$"1*HLWMK@1&!#;7$$"1f=PuK(=o)!#;$"1UfO$)f7GT!#;7$$"1e>RysV&**)!#;$"2ae))4z-Y6$!#<7$$"1W)oP&px&H*!#;$"2/yx&[@Ei@!#<7$$"1Z)pRf&Q&f*!#;$"2l;Z)*\9-B"!#<7$$"1)\**)zzK3**!#;$"1mVGiVceF!#<7$$"2U&3<Mfl>5!#;$!2kOVknW"ee!#=7$$"117CoZ<\5!#:$!2#=/#*o91^9!#<7$$"2MrU&oqw"3"!#;$!0s+8_`hQ#!#:7$$"23<Mosw76"!#;$!1ben0nZ9K!#;7$$"2u\**)zeXT6!#;$!2E^.utyN/%!#<7$$"2-.17IZ@<"!#;$!21qbI5s!o[!#<7$$"1W([(*oe.?"!#:$!1#)R)okC$4c!#;7$$"2(4>Q;6VI7!#;$!1ir[(\[>Q'!#;7$$"2Pu[(HYFg7!#;$!1aQbj+"38(!#;7$$"19Gc7z>#H"!#:$!1oP#e<T@"z!#;7$$"1oOtE)o.K"!#:$!1uHe]Hq%e)!#;7$$"28MoO@sGN"!#;$!1"Ge#=%y4M*!#;7$$"2<OsW,/AQ"!#;$!2eHQZ8K0+"!#;7$$"2Z&4>)HE7T"!#;$!2#G!\i#=dk5!#;7$$"1<MowXVU9!#:$!2o@0aPb:8"!#;7$$"2vZ&4*4pPZ"!#;$!1#pG!Q,&o>"!#:7$$"2;Pu[()yA]"!#;$!2)\!ykb_XD"!#;7$$"1f<N!*\bK:!#:$!2[r1L:]SJ"!#;7$$"2,+++_,@c"!#;$!0p87WY.P"!#97$$"2fC\)p-8%f"!#;$!2.g'*>.+%H9!#;7$$"2`3<M/W=i"!#;$!2d%z"f6U)y9!#;7$$"2s\**)>Vn`;!#;$!2AU-K%GtL:!#;7$$"1He;`kf$o"!#:$!2le;([_[$e"!#;7$$"1"Gc7dHKr"!#:$!2^UaCH#*4j"!#;7$$"2xc8FE')Gu"!#;$!2<<7))yynn"!#;7$$"2*oPv5U8t<!#;$!2PXBW7m;s"!#;7$$"2KsW*Gl>0=!#;$!2#R&4m8[sw"!#;7$$"1x`2NXfM=!#:$!1Y%[scNs!=!#:7$$"005?99O'=!#9$!2[R](RI,X=!#;7$$"106ACcj%*=!#:$!2(Hy`sR`$)=!#;7$$"2W)oPv&*eD>!#;$!2;H%3kE0?>!#;7$$"2$=Os%esL&>!#;$!1j#o)o%)>^>!#:7$$"2.29Gc$G')>!#;$!2M*\;Aa4')>!#;7$$"2xd:J%>#Q,#!#;$!1'3/q*3j8?!#:7$$"28Ig?<Jg/#!#;$!1c!fKI7R/#!#:7$$"0-/3wrn2#!#9$!02Zv&y(32#!#97$$"1BY#\W>U5#!#:$!1FUA_xN$4#!#:7$$"2'4>QOs3N@!#;$!2xg[\nQo6#!#;7$$"2'*)zfR'fh;#!#;$!2dlmPh]&Q@!#;7$$"2v_06O%3(>#!#;$!1aJr:@Ce@!#:7$$"2X"HeObvDA!#;$!2X2&Gu**yu@!#;7$$"2BX!4yGWbA!#;$!2&=L79=>!>#!#;7$$"1Gd9pb9'G#!#:$!1&*RS#GmU?#!#:7$$"2oS"G'z\nJ#!#;$!1%4PGO>k@#!#:7$$"229Gc+I#[B!#;$!2cO3)oe'pA#!#;7$$"2)yd:"zdfP#!#;$!2l/D/`8YB#!#;7$$"1f<N5F<2C!#:$!2;c'H%4$QTA!#;7$$"1D]+,e^QC!#:$!2e%RF$p>iC#!#;7$$"2W"Hew1soC!#;$!0dERg@!\A!#97$$"1KkG(f\h\#!#:$!1W*)Ru^)*\A!#:7$$"2X#\)pBl(GD!#;$!1,79hD<\A!#:7$$"005?'\RcD!#9$!2Y))f#3'>oC#!#;7$$"1:Ig+\`)e#!#:$!1:_Pr:;UA!#:7$$"2P!3;#>%)ph#!#;$!2P[%y)p9jB#!#;7$$"1\)pRd(>[E!#:$!28Xp!zu.GA!#;7$$"1pQx%4?zn#!#:$!2ylv)RWM=A!#;7$$"2$yc8FC$*3F!#;$!1&>"*3CZj?#!#:7$$"28BY#H9TPF!#;$!2;EjK"ej$>#!#;7$$"0,-/))H"oF!#9$!2UgNsO1"y@!#;7$$"1Z%*)yrP+!G!#:$!1$pVyOx*f@!#:7$$"2c5@Ue8y#G!#;$!2E6+SDQD9#!#;7$$"2'>S!3G7y&G!#;$!2X#eqr.(>7#!#;7$$"1V([(pQ!)))G!#:$!1-%o3bJ))4#!#:7$$"1KjE$4B">H!#:$!2;'3pKeLu?!#;7$$"1D]+,&e%[H!#:$!2jKwG(\))[?!#;7$$"2Nu[(p/.")H!#;$!1.L<(o4'=?!#:7$$"2%3=OKyH5I!#;$!1RoGAhf*)>!#:7$$"1BZ%*))paTI!#:$!2V!\weosc>!#;7$$"2MtY$pL')pI!#;$!2"*yw-ub_#>!#;7$$"29JiCl?35$!#;$!2;wuNa9!*)=!#;7$$"1$e;L/\*HJ!#:$!27_CG?kJ&=!#;7$$"2$=Os/mRgJ!#;$!2%=&pWKwQ"=!#;7$$"2C`18mk,>$!#;$!0k$GSFnt<!#97$$"1X*)yd!H8A$!#:$!2WJ*Q!R%oH<!#;7$$"2$*)ydvSM^K!#;$!2#3S4!3#[&o"!#;7$$"2/;KkwQ?G$!#;$!1jszn`TQ;!#:7$$"1W([(*GzCJ$!#:$!2mPHPSx)*e"!#;7$$"20BY#p4XSL!#;$!1sIHo@kV:!#:7$$"2Nw_0n4DP$!#;$!13>zuos)["!#:7$$"2%ze<bL=,M!#;$!2j%RNg&oyV"!#;7$$"2we<N[b<V$!#;$!2j!\$*=<$=Q"!#;7$$"1#Gc7"p,hM!#:$!2UVzi\YkK"!#;7$$"2d:Ji'z]$\$!#;$!2NBN5#>%HE"!#;7$$"2:JiC$Gk@N!#;$!2iQyG#fC17!#;7$$"2CV'Gdhd`N!#;$!2Tb(z!Gx*R6!#;7$$"1/4=;*zEe$!#:$!21^)4*>/y2"!#;7$$"1kGda[^9O!#:$!2wU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ratint(1/(x^2+b*x+c),x);

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkkYWJzR0YmNiMsJkkieEdGKCIiIiEiIkYvI0YwIiIkLUYkNiMtRis2IywmRi5GLyEiJUYvI0YvRjI=

unassign('b','c'); _________________________________________________________________________________Exercise 5Terms like 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 in 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 are associated with places where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4= blows up to infinity (whichare known as the poles of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4=). More precisely, if there is a term 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 in the integral, thenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4= must blow up at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ1RidGL0YyRj5GK0Y+. Conversely, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4= blows up at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ1RidGL0YyRj5GK0Y+ then there will usually be a 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 in the integral, but not always; there may instead be terms of the form 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. Thefunctions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSVtc3ViR0YkNiUtRiw2JVEiZ0YnRi9GMi1GIzYkLUkjbW5HRiQ2JFEiN0YnL0YzUSdub3JtYWxGJ0ZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmduLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkZDRkNGQ0YrRkM= are examples of this. Here we consider 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; there are poles at 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GK0Y+, and corresponding terms 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and 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 in the integral. y := g[2](x); z := ratint(y,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqJCwmSSJ4R0YoIiIiISIiRjJGM0YyKiQsJiokRjEiIiNGMkYzRjJGM0YyKiQsJiokRjEiIiRGMkYzRjJGM0YyNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ6R0YoLCotSSNsbkdGJTYjLUkkYWJzR0YmNiMsJkkieEdGKCIiIiEiIkY3IyIjNiIiJy1GMDYjLUYzNiMsJkY2RjdGN0Y3I0Y4IiIjLUYwNiMsKCokRjZGQkY3RjZGN0Y3RjcjRjhGOyomIiIkI0Y3RkItSSdhcmN0YW5HRiU2IywmKiZGSUZKRjZGNyNGQkZJKiRGSUZKI0Y3RklGNyNGOEZJNyNGLg==

plot([y,z],x=-5..5,-10..10);

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Here we consider 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. There are no LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEjbG5GJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JKG1mZW5jZWRHRiQ2JC1GIzYkRitGPEY8RjxGK0Y8 terms in the integral, but nonetheless there are polesat LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GK0Y+, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjJGJ0Y+Rj5GK0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GK0Y+. y := g[3](x); z := ratint(y,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqJCwmSSJ4R0YoIiIiISIiRjIhIiNGMiokLCZGMUYyRjRGMiEiJEYyKiQsJkYxRjJGN0YyISIlRjI3I0Yu

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Here we consider 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. There is a pole at 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, and a corresponding term 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 in theintegral. y := g[4](x); z := ratint(y,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqJCwmSSJ4R0YoIiIiRjJGMiEiIkYyKiQsJiokRjEiIiNGMkYyRjJGM0YyKiQsJiokRjEiIiRGMkYyRjJGM0YyNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ6R0YoLCotSSNsbkdGJTYjLUkkYWJzR0YmNiMsJkkieEdGKCIiIkY3RjcjIiIlIiIkLUknYXJjdGFuR0YlNiNGNkY3LUYwNiMsKCokRjYiIiNGN0Y2ISIiRjdGNyNGQyIiJyomRjojRjdGQi1GPDYjLCYqJkY6RkdGNkY3I0ZCRjoqJEY6RkcjRkNGOkY3I0Y3Rjo3I0Yu

plot([y,z],x=-5..5,-10..10);

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_________________________________________________________________________________Exercise 6We only get terms like 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 or 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 in 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 if the denominator of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiZ0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY+Rj5GPkYrRj4= has repeated roots.A repeated root at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ1RidGL0YyRj5GK0Y+ gives a factor 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 in the denominator, with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPEYnRjsvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUi1GLDYlUSJuRidGL0YyRjtGK0Y7. This gives a term 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 in the integral (and possibly also some terms like 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 for smaller values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=).Here we consider 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: y := g[3](x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqJCwmSSJ4R0YoIiIiISIiRjIhIiNGMiokLCZGMUYyRjRGMiEiJEYyKiQsJkYxRjJGN0YyISIlRjI3I0Yu

The denominator has repeated roots at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+Rj5GK0Y+, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjJGJ0Y+Rj5GK0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GK0Y+: d := denom(factor(y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoKigsJkkieEdGKCIiIiEiIkYxIiIjLCZGMEYxISIjRjEiIiQsJkYwRjEhIiRGMSIiJTcjRi4=

These show up in the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as points where the graph touches the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis without crossing it, or where it crosses but the tangent line at the crossing point is horizontal: plot(d,x=0..4,-0.3...0.3);

6*-%'CURVESG6$7dw7$$""!!""$!%!['!""7$$"29D]+,UI5#!#=$!1QFe/s^\e!#87$$"2EX!4=w)G$R!#=$!1')R&3<8ZM&!#87$$"1[&4>QS2*f!#<$!2E=dNx#GA[!#97$$"1U#['HRBi!)!#<$!2k['45_STV!#97$$"2`2:I?)Q75!#<$!1JL)p"eN/R!#87$$"2:KkGPHN?"!#<$!2'He/,@[LN!#97$$"2Gd9H)\W,9!#<$!1-]R6<u"=$!#87$$"2)[(\**eHhg"!#<$!1QV$*3I)*\G!#87$$"28Fa3xd,"=!#<$!1HHnTmA\D!#87$$"2%He;Le-??!#<$!2Zu#GzwgoA!#97$$"1<MoOx([?#!#;$!2ACt2CGP/#!#97$$"0&**)z>xHT#!#:$!2(HTrA**e8=!#97$$"1#Ryc8J>i#!#;$!1yV-"o'40;!#87$$"1AV'G(pHBG!#;$!2wf#>^W#QU"!#97$$"2`5@UWch+$!#<$!2\/4EavXF"!#97$$"2a06A/%fBK!#<$!2iW[(>Lc96!#97$$"1E_/4Az2M!#;$!1$*G'ow-k#**!#97$$"2&e;Lm%e?i$!#<$!0@MUJB<l)!#87$$"1^-05/s6Q!#;$!1R8n(>X8k(!#97$$"1`06A'4)>S!#;$!1[%Q1IH)[m!#97$$"2/-/3wfz@%!#<$!1$o6q(\e1e!#97$$"1"3;Kk3ZU%!#;$!0c`c#R)[-&!#87$$"2DV'Gd'oXh%!#<$!2&*R8^s*R'Q%!#:7$$"2;He;Le$>[!#<$!0#)y^=$*[x$!#87$$"117C[s2K]!#;$!1MksWXN;K!#97$$"1hAX!p\s@&!#;$!2#e_95u_(y#!#:7$$"1D]+,5C<a!#;$!2$oURp(e%yB!#:7$$"1\)pRf^Qi&!#;$!2MpbHfj*3?!#:7$$"1pPv](zf#e!#;$!1Xc=1XF%p"!#97$$"1#Ryc8\:-'!#;$!2E*3pW))=H9!#:7$$"0&)pRf&pQi!#:$!2/H(>BzEv6!#:7$$"1=OsW8"QV'!#;$!1Crort`$z*!#:7$$"1$Qw_0R@k'!#;$!1p\wjJg**z!#:7$$"1:JiC\"4$o!#;$!1:&3DQB&4m!#:7$$"1-.17oHPq!#;$!1f"*R%)Rj:`!#:7$$"06AW3'[Js!#:$!1-&**po%)zG%!#:7$$"1zc8F)pWV(!#;$!2bf"p!*p$fQ$!#;7$$"1U%)oPN#Hj(!#;$!2`**ox-WDl#!#;7$$"1BX!4='oSy!#;$!22"[()H@3@?!#;7$$"1)[(\*H'yS!)!#;$!2'***yI.Zm_"!#;7$$"1(Hf=d<aC)!#;$!2k$)f(p<))>6!#;7$$"0&)pRRa$[%)!#:$!0ZqrQ?k+)!#:7$$"1KkGdA$[j)!#;$!1Ay7?dl+d!#;7$$"1<LmKpb[))!#;$!1;)z&*==*zO!#;7$$"1`2:I[rR!*!#;$!2_RI(Q\WVB!#<7$$"1ya4>!HNC*!#;$!2<<u.tG>K"!#<7$$"1,,-/_gQ%*!#;$!1>`Ju)ogj'!#<7$$"1Jf=PA@b'*!#;$!2yW\')=gYD#!#=7$$"1,.17!yF%)*!#;$!2XG$Qr&ejF%!#>7$$"206AWomb+"!#;$!1M-]^)y;#[!#>7$$"2)e<N!>p\-"!#;$!1#)ov*z_Mz)!#=7$$"2`18E[#>Y5!#;$!1v'=h1i!)p#!#<7$$"2U'Gd%f/X1"!#;$!2a=d:(o^!y%!#=7$$"2T"Gc_*f`3"!#;$!1Z"p*z#o@\(!#<7$$"0(Rz)>0b5"!#9$!2&G;/zRDE5!#<7$$"2d6BYGPc7"!#;$!1KSI>kM-8!#;7$$"2NqS"3`pX6!#;$!1voX&>#yk:!#;7$$"2OtY$*yk\;"!#;$!2etU!Hbp'z"!#<7$$"27AW)3cz&="!#;$!2K$z%f.T%=?!#<7$$"2jD^-*)[c?"!#;$!2Ks"o,\X(>#!#<7$$"2>NqS`VlA"!#;$!200ta]a!\B!#<7$$"2tX"H=#eaC"!#;$!10qwEty_C!#;7$$"218E_[ijE"!#;$!2'[2ByAgID!#<7$$"2jD^-8&Q'G"!#;$!2m;g`=t+d#!#<7$$"0*zfR!fjI"!#9$!0(3b()p@xD!#:7$$"2KjE`'=AF8!#;$!2&f)=!Hk,`D!#<7$$"2jBZ%*GPkM"!#;$!1)[SUI>\]#!#;7$$"1/3;7l6m8!#:$!2'RX,/90LC!#<7$$"2#4=O7Z%yQ"!#;$!2#GlcxO#4L#!#<7$$"2sW*)y"y^29!#;$!2/Vh&e@N@A!#<7$$"2<LmKDPwU"!#;$!2([<Qfo#f4#!#<7$$"2--/3?)4[9!#;$!2N$)=3?#*y&>!#<7$$"2%He;$z0pY"!#;$!1C"H@03X#=!#;7$$"2(RzexS&p["!#;$!2YuG)QlBy;!#<7$$"2"He;`(\o]"!#;$!1p)QaJw9`"!#;7$$"2Ga3<%>8G:!#;$!2Por_=y`P"!#<7$$"2*yd:^D"pa"!#;$!1/PwDKOS7!#;7$$"2X*)ydZ"eo:!#;$!1^%\cy.)*3"!#;7$$"1T#['4g8)e"!#:$!/BZ`:f-'*!#:7$$"2*pRz)>%[2;!#;$!1lce.^a"R)!#<7$$"2Y%*)y<(*GG;!#;$!1sX\M`mxr!#<7$$"1&)pR*Rz"\;!#:$!1;b]YQkeg!#<7$$"2X"He;f=o;!#;$!1AIlK#3;8&!#<7$$"2$Ryc$**p$)o"!#;$!144\k%4VC%!#<7$$"*on!3<!")$!13QVuD+vM!#<7$$"2v\**)z,UH<!#;$!2kF]w_Upu#!#=7$$"2OsW*Gg*yu"!#;$!1'z5(p1%G?#!#<7$$"2;LmK@;"p<!#;$!1d#\FZ:7n"!#<7$$"2Gb5@Ik!*y"!#;$!2/7gW(4\c7!#=7$$"2U&3<9(>)3=!#;$!0:$3[FG.#*!#<7$$"2`/4=%3fG=!#;$!1)H/6fO0^'!#=7$$"2E^-09c([=!#;$!1["=C.EzP%!#=7$$"2C['H>58q=!#;$!2'fmi<1q-F!#>7$$"27D]+pH(*)=!#;$!1En/Iy*Gh"!#=7$$"2nOtY4w!4>!#;$!1V<:)Hz&)z)!#>7$$"2-29G3d(H>!#;$!1&\A^vb2$R!#>7$$"2jD^-0$R]>!#;$!1"[T]8h#Q8!#>7$$"2BT#[c]"*o>!#;$!1fB%))\Fr=$!#?7$$"2PrU&3d&3*>!#;$!0-DO]cay(!#@7$$"2&=PuGY@4?!#;$"1f`DECDzw!#A7$$"1,-/[uoI?!#:$"2$RRE"[R,r#!#@7$$"2mMpQ<"=^?!#;$"2'3OL`sl+7!#?7$$"2([(\*H'z%p?!#;$"1H$*y@J@wG!#>7$$"0%zed"e+4#!#9$"1.fGTM3]f!#>7$$"2mKlI4t26#!#;$"2<#)G8q&e[5!#>7$$"2<NqSd*QJ@!#;$"2UX">q,v_;!#>7$$"1w_0"p.0:#!#:$"2aU,bBw*\B!#>7$$"1-.1__Hq@!#:$"2v,/\/ya?$!#>7$$"1>Qw7Pw!>#!#:$"1YL3v$Q7A%!#=7$$"1rU&3`m6@#!#:$"20v!)o30%[`!#>7$$"1Fa3PL:KA!#:$"1hE'HZnJg'!#=7$$"1'=Pu_Q1D#!#:$"1j]CDdblx!#=7$$"1Ryct%[9F#!#:$"1)Q2"\>j4"*!#=7$$"2OoOt'QM#H#!#;$"2#G*4'*3rk/"!#=7$$"2mFb5X![7B!#;$"2VF-ZhTV<"!#=7$$"2X&4>)Rm2L#!#;$"2#>!=vw>bG"!#=7$$"2(\**)z:5DN#!#;$"2$*R$QTeS39!#=7$$"2nOtY(*H4P#!#;$"02j8&>6-:!#;7$$"2.,-/gcBR#!#;$"116pX**Q'f"!#<7$$"2'pQx%zA8T#!#;$"2wGHv7+Ym"!#=7$$"2%**)zfrJ@V#!#;$"2v%*)\b[4@<!#=7$$"2jC\)Hn%>X#!#;$"28OW#[drb<!#=7$$"2BX!4=;isC!#;$"2&\(oW"[(4x"!#=7$$"2x[(\>wg"\#!#;$"1Gm_$yzew"!#<7$$"2LnMpe'37D!#;$"1&3#p_7-S<!#<7$$"1lHfy%eL`#!#:$"1Wr]1a_"p"!#<7$$"2229Gsv=b#!#;$"2G;Ft)*=Cj"!#=7$$"2mMpQ&[(=d#!#;$"2ETd.BlCb"!#=7$$"1He;8f`#f#!#:$"2dJ@@b'Qa9!#=7$$"28AW)G([Fh#!#;$"2oSlZ'=rX8!#=7$$"2MoOtm0Bj#!#;$"2jgv>Y47B"!#=7$$"2$He;8.-aE!#;$"2_H#R@Ti'4"!#=7$$"117C))=`tE!#:$"0(pPc!z3s*!#<7$$"2D['HfYO%p#!#;$"1VsYEbp'Q)!#=7$$"2c:JiCUKr#!#;$"1XGXt^F,s!#=7$$"1u[(\V!)Qt#!#:$"10?j5%f'ff!#=7$$"2b06AO*H`F!#;$"1KH$Qz)Qn[!#=7$$"17C[OxftF!#:$"2Z?BKz)HEQ!#>7$$"2%)oPv5VMz#!#;$"1dJmi,qCH!#=7$$"2nHf=P>U"G!#;$"1<_5U#Qk6#!#=7$$"1$f=PQHU$G!#:$"19BJ^l+v9!#=7$$"2Oxa4^#paG!#;$"1Ftm0xqu&*!#>7$$"2%He;$>')\(G!#;$"1sOvD%p@v&!#>7$$"1([(\zRj$*G!#:$"1T8?f(QbF$!#>7$$"2b<NqW1]"H!#;$"2oJ*zOt2m9!#?7$$"0#RyOA7MH!#9$"0,>q!4%Hu&!#>7$$"1#Rycl.X&H!#:$"2or5B@]LU"!#@7$$"2W&3<u7,uH!#;$"2$zh!f](fU;!#A7$$"1Pu[x>n&*H!#:$"2uo455%Qz8!#D7$$"1u[(\bGW,$!#:$"2;A="[U%f$=!#B7$$"2Y&4>QurNI!#;$"1>rK^)>M\(!#@7$$"1.17W*>^0$!#:$"1-WuWoYzX!#?7$$"14>QOKMwI!#:$"2'e0!)p^8E=!#?7$$"2(3<M[`l%4$!#;$"1Xe?Dw&*>Y!#>7$$"1e;L12^:J!#:$"23+`=6jf5"!#>7$$"2V"Gc_flNJ!#;$"1>%QGYUBE#!#=7$$"0'>RQ!)ybJ!#9$"1l#["R?_EU!#=7$$"2xa4>1Ye<$!#;$"1V$G)G3Lft!#=7$$"1yb6Vb6&>$!#:$"2&)>#\)*y4#>"!#=7$$"2b18EOYf@$!#;$"2OuW=#Gv>>!#=7$$"205?Sk*zNK!#;$"1#H#o*3Qm"H!#<7$$"1'>RyG%pcK!#:$"2Y#z/3#H$)Q%!#=7$$"2<Ig?(*3cF$!#;$"1*4a(z;&=?'!#<7$$"1v\**QK^'H$!#:$"0EuwJ[Z)))!#;7$$"225?S)e`;L!#;$"2\8tibG$H7!#<7$$"2W$oO$z4lL$!#;$"2NDQa2a7n"!#<7$$"2uZ&4>EPdL!#;$"2sjF<sLpE#!#<7$$"2/3;K/)ewL!#;$"1v4Z`,PjH!#;7$$"2%['HfEniR$!#;$"1py%=?`T&Q!#;7$$"1`18ma*zT$!#:$"18#*Q6\())3&!#;7$$"2=He;dowV$!#;$"11QVmo)*yk!#;7$$"1w^.2!)ydM!#:$"1"*GDj5?>#)!#;7$$"2X'Gda*[#yM!#;$"20]5Rx$)y."!#;7$$"2PnMpacq\$!#;$"07/$y2,x7!#97$$"1%yc8$[5<N!#:$"2(*\ASSW=e"!#;7$$"2PnMp]+q`$!#;$"1o<On()RV>!#:7$$"1(Qxap#GeN!#:$"0%\[")4w0C!#97$$"1BY#\Ijqd$!#:$"2viC_[E'))G!#;7$$"1RxaHAt)f$!#:$"2c$[(p_Vja$!#;7$$"2a3<Mw'G=O!#;$"1;4,T5"eC%!#:7$$"19Gc_\jPO!#:$"1#G:FE/40&!#:7$$"2%*yd:ZS%eO!#;$"10=IA.Efg!#:7$$"1He;`,LzO!#:$"1&Q<,Fi2C(!#:7$$"1f<NqmL)p$!#:$"1)\/_\$[#[)!#:7$$"2NoOtu?&=P!#;$"1d6y%)fR(***!#:7$$"1W)oPV=#QP!#:$"2[<[j;p%p6!#:7$$"1U$oO$4dfP!#:$"1[x5kj%4Q"!#97$$"2vc8FyY!yP!#;$"2Z$*R;yf(*e"!#:7$$"2d<Nq'pE*z$!#;$"2oY5q++F'=!#:7$$"2uRze0:#>Q!#;$"2;Rqi#GEb@!#:7$$"2))pRzYq*QQ!#;$"2xEOR8?L[#!#:7$$"2%*)yd&fT(eQ!#;$"2(zyAg*=S&G!#:7$$"2nNrU*o!*yQ!#;$"1/Ai'f=0G$!#97$$"2lKlIx"G+R!#;$"1,n%>ZC=z$!#97$$"2d4>QW!))>R!#;$"1vfQ*>X*>V!#97$$"28@U%[oARR!#;$"1Wqu-do-\!#97$$"1:HeOy!*fR!#:$"1oZa0b"**f&!#97$$"/,-/Qa!)R!#8$"1%oXC*z()zj!#97$$"#S!""$"$?(!""-%&COLORG6'%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-%+_ATTRIBUTEG6#/%'sourceG%,mathdefaultG-%%VIEWG6$;$""!!""$"#S!"";$!"$!""$""$!""-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"x6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"G6"

The integral has terms of the form 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 for 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. ratint(y,x);

LCgqJCwmSSJ4RzYiIiIiISIkRidGKCMhIiIiIiQqJCwmRiVGJyEiI0YnRi4jRioiIiMqJCwmRiVGJ0YqRidGKkYq

Here we consider 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: y := g[4](x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoLCgqJCwmSSJ4R0YoIiIiRjJGMiEiIkYyKiQsJiokRjEiIiNGMkYyRjJGM0YyKiQsJiokRjEiIiRGMkYyRjJGM0YyNyNGLg==

The denominator has a root at 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, which is not repeated: d := denom(factor(y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoKigsJiokSSJ4R0YoIiIjIiIiRjNGM0YzLCZGMUYzRjNGM0YzLChGMEYzRjEhIiJGM0YzRjM3I0Yu

This is visible in the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Note that where the graph meets the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis, the tangent line is nothorizontal, corresponding to the fact that the root is not repeated. plot(d,x=-2..2,-10..10);

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The integral has no terms of the form 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. ratint(y,x);

LCotSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkkYWJzR0YmNiMsJkkieEdGKCIiIkYvRi8jIiIlIiIkLUknYXJjdGFuR0YlNiNGLkYvLUYkNiMsKCokRi4iIiNGL0YuISIiRi9GLyNGOyIiJyomRjIjRi9GOi1GNDYjLCYqJkYyRj9GLkYvI0Y6RjIqJEYyRj8jRjtGMkYvI0YvRjI=

Here we consider 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: y := g[6](x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ5R0YoKixJInhHRigiIiIsJkYvRjBGMEYwISIjLCZGL0YwIiIjRjBGMCwmRi9GMCIiJEYwRjIsJkYvRjAiIiVGMEYwNyNGLg==

The denominator has repeated roots at 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 and 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: d := denom(factor(y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoKiYsJkkieEdGKCIiIkYxRjEiIiMsJkYwRjEiIiRGMUYyNyNGLg==

These visible in the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, as points where the curve meets the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis without crossing it.Note that the tangent line is horizontal at these points. plot(d,x=-4..2,-10..10);

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The integral has terms of the form 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 for 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 and 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. ratint(y,x);

LCoqJCwmSSJ4RzYiIiIiIiIkRichIiIjISIkIiIlLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2Iy1JJGFic0dGMDYjRiQjRiciIiMqJCwmRiVGJ0YnRidGKSNGKEYsLUYuNiMtRjQ2I0Y5RjY=

We now integrate some random rational functions, and observe that we do not get any terms 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. The reason is simply that the denominator is a random polynomial, which will almostnever have repeated roots. The roots of a random polynomial are randomly distributed, so it would be a big coincidence for two roots to be in the same place, and this does not often happen. r := randpoly(x)/randpoly(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoKiYsLiokSSJ4R0YoIiImISIoKiRGMSIiJSIjQSokRjEiIiQhI2IqJEYxIiIjISMlKkYxIiMoKSEjYyIiIkY/LCxGNCEjaUY3IiMoKkY6ISN0RjEhIiUhIyQpRj8hIiI3I0Yu

ratint(r,x);

LC5JInhHNiIkISs+eio+eSIhIzUtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJEYjIiIjIiIiJCIrdlssKEcnRidGMUYjJCIrdiJIWSlwRickIStTMTBzOkYnLUknYXJjdGFuR0YqNiMsJkYjJCIrPSd5WlMiISIqJCIrZSlHZiFcRidGMSQiK20lZitiKkYnLUYpNiMsKEYvRjEkIis5W0tIQEY+RjFGIyQhK1khekhFI0Y+JCIrXispKVtSRictRjk2IywmRiMkIis9XEQmMyJGPiQhK3FYJnpBIkY+RjEkIisjZTRiWClGJ0YvJCIrIUhoXmsmISM2

r := randpoly(x)/randpoly(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoKiYsLiokSSJ4R0YoIiImISM1KiRGMSIiJSIjaSokRjEiIiQhIyMpKiRGMSIiIyIjISlGMSEjVyIjciIiIkY/LC5GMCEjPEY0ISN2RjdGM0Y6ISIoRjEhI1MiI1VGPyEiIjcjRi4=

ratint(r,x);

LC5JInhHNiIkIitUSE4jKWUhIzUtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJEYjIiIjIiIiJCIrQkhAdiMpRidGMUYjJCErUiVlYW4kRickIisjZnU9MSJGJy1JJ2FyY3RhbkdGKjYjLCZGIyQiKzZ6VUE2ISIqJCErbyY9RjEjRidGMSQiK0xEJT1FJkYnLUYpNiMtSSRhYnNHRis2IywmRiNGMSQhK3d0PGtoRidGMSQhK0olRyJHV0YnLUYpNiMtRkY2IywmRiNGMSQiKyN6XSRlVUY+RjEkIStARVYjKnpGPi1GKTYjLUZGNiMsJkYjRjEkIislXHh0OCJGPkYxJCIrNWdsISk+Rj4=

r := randpoly(x)/randpoly(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJyR0YoKiYsLiokSSJ4R0YoIiImISNdKiRGMSIiJSIjQiokRjEiIiQiI3YqJEYxIiIjISMjKkYxIiInIiN1IiIiRj8sLkYwIiNzRjQiI1BGNyEjQkY6IiMoKUYxIiNXIiNIRj8hIiI3I0Yu

ratint(r,x);

LC5JInhHNiIkIStXV1dXcCEjNS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsKCokRiMiIiMiIiIkIiskKTNdVjYhIipGMUYjJCErKHkmNDk4RjQkIStqPGJRIiohIzctSSdhcmN0YW5HRio2IywmRiMkIitWJCpHJj0iRjQkIStoJz16eShGJ0YxJCErMkxPS0khIzYtRik2IywoRi9GMSQiK1BBMXNFRidGMUYjJCIrZUYieTQmRickIis5WEZTTUYnLUY7NiMsJkYjJCIreG5tQkFGNCQiK0olPXptJkYnRjEkIitEXSNIYSJGNC1GKTYjLUkkYWJzR0YrNiMsJkYjRjEkIisrTT89OEY0RjEkIitKIkhSYCdGOQ==