Question

integrate cos^2(2x) sin(2x)

Answer 1

u mean
\[ \large \int\cos^2(2x)\ \sin(2x)\,dx \]?

Answer 2

use the substitution
\[ \large u=\cos(2x) \]

Answer 3

I get -(1/6)cos^3(2x)

Answer 4

let's see
\[ \large u=\cos(2x)\qquad\text{then}\quad du=-2\sin(2x) \]

Answer 5

so
\[ \large \int\cos^2(2x)\sin(2x)\,dx=\int u^2\frac{du}{-2} \]
\[ \large =\frac{1}{-2}\frac{u^3}{3} \]
so
\[ \large \int\cos^2(2x)\sin(2x)\,dx=-\frac{1}{6}\cos^3(2x)+C \]