Trigonometric Integration problem

Integrate x[sin^5(x^2)](cosx^2) dx

Do you apply one of the trig transformations using the identity cos^2(x)+sin^2(x) = 1?

If so, how?

I tried typing the problem at wolfram and it says that computation has timed out... :(

Hello,

let's wrtie the integral as:

∫ sin⁵(x²) x cos(x²) dx =

let:

sin(x²) = u

(differentiating both sides)

d[sin(x²)] = du

2x cos(x²) dx = du

x cos(x²) dx = (1/2) du

then, substituting:

∫ sin⁵(x²) x cos(x²) dx = ∫ u⁵ (1/2) du =

(pulling the constant out)

(1/2) ∫ u⁵ du =

(1/2) [1/(5+1)] u^(5+1) + C =

(1/2)(1/6)u⁶ + C =

(1/12)u⁶ + C =

let's substitute back sin(x²) for u, ending with:


(1/12)sin⁶(x²) + C


I hope it helps