Math Homework help!! 10 points!?
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Math Homework help!! 10 points!?

[From: ] [author: ] [Date: 12-06-01] [Hit: ]
......
Find the following integrals:

cosec^2x / (2 + cotx)^3 . dx

the answer starts with

let y = (2 + cotx)^ -2

my question is how did they know that y should be equal to (2 + cotx )^ -2

thankyou.

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I will answer your question first

They know that the substitution must be (2+cotx)^(-2) because when this is differentiated it will generate a power of -3. Since you see a power of +3 in the integral it means that there is probably some canceling out as we will see!

y=(2+cotx)^(-2) dy=(-2)(2+cotx)^(-3)(-1/sin^2x)dx (the term -1/sin^2x comes from differentiating

cotx)

Substitute these in the original integral bearing in mind that since y=(2+cotx)^(-2) you sqyare both sides to get :

to get y^2=2+cotx

Integral((( sin^2x)sin^2x)/(y^6)*2*(2+cotx)^(-3))dy=…

and going back to the x 1/2(2+cotx)^(-2)+C
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