evaluate the integral?

Question

integral of x*(cos(x^2))^5

Can you show me how to do it?? or give me a clue?

Would that be easier to do with the method of integral by part? or tri-substitution?

Answer 1

substitution:
let u = x² ... du = 2 dx

∫1/2(cosu)^5 du
∫1/2(cosu)^4 * cos u du
∫1/2(1-sin²u)² cos u du

then let w = sin u ... dw = cos u du
∫1/2 (1-w²)² dw

I guess you can finish from here. Just return the variables from w to u then to x.

d:

Answer 2

Its been a while, but I think that integration by parts should do it ... however, be warned: it will still take four or five steps.
I think that your factor on the cos(x^2) should reduce by one each step.
Good luck

Answer 3

integral of x*((cos(x^2))^5) dx =
(5*sin(x^2)/16) + (5/96)*sin(3x^2) + (1/160)*sin(5x^2)
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Answer 4

Why don't you try both ways, what a better way to learn?