Integration Examples

from Chapter 8

>	with(plots):

Warning, the name changecoords has been redefined

Example 1

>	Int(ln(2*x)/x^2,x);

>

value(%);

Example 2

>	Int(x*exp(-x/2),x=0..4);

>

value(%);

>

evalf(%);

>	plot(x*exp(-x/2),x=0..4,filled=true);

Example 3

>	Int(exp(x)*sin(x),x);

>

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>	Int(exp(x)*sin(x),x=0..1);

>

value(%);

>

evalf(%);

>	plot(exp(x)*sin(x),x=0..1,filled=true);

Example 4

>	Int(((sin(2*x))^5)*cos(2*x),x);

>

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Example 5

>	Int((cos(x/3))^3,x);

>

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Note that the answer given above is equivalent to the answer given on my web site.

Example 6

>	Int((sec(x))^3*tan(x),x);

>

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Example 7

>	Int(sin(-4*x)*cos(3*x),x);

>

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Example 8

>	Int(4/sqrt(4-x^2),x);

>

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Example 9

>	Int(x/sqrt(9-x^2),x);

>

value(%);

The picture used on my web site is drawn below.

>	Triangle:=plot([[0,0],[1,1],[1,0],[0,0]],axes=none,thickness=3):

>	Theta:=textplot([.2,.09,'q'],font=[SYMBOL,24]):

>	Hyp:=textplot([.43,.56,'3'],font=[HELVETICA,BOLD,24]):

>	Opp:=textplot([.9,.5,'x'],font=[HELVETICA,BOLD,24]):

>	Adj:=textplot([.6,.1,'sqrt(9-x^2)'],font=[HELVETICA,BOLD,24]):

>	display(Triangle,Theta,Hyp,Opp,Adj);

Example 10

>	Int(x/sqrt(x^2-6*x+5),x);

>

value(%);

The picture used on my web site is drawn below.

>	Hyp:=textplot([.43,.56,'x-3'],font=[HELVETICA,BOLD,24]):

>	Opp:=textplot([.9,.5,'sqrt(x^2-6*x+5)'],font=[HELVETICA,BOLD,24]):

>	Adj:=textplot([.6,.1,'2'],font=[HELVETICA,BOLD,24]):

>	display(Triangle,Theta,Hyp,Opp,Adj);

Example 11

>	Int(1/(1-t^2)^(5/2),t=0..sqrt(3)/2);

>

value(%);

>	plot(1/(1-t^2)^(5/2),t=0..sqrt(3)/2,labels=[t,y],filled=true);

>