Write $\cos^8(x)$ in terms of $\sin(x)$?

Question

I am currently working on an integral using u substitution and have hit a wall at the above point. The integral in question is $\int$$\sin^7(x)$$\cos^9(x)$ using $u=sin(x)$.

After finding $du=cos(x) dx$, I am asked to express $\cos^8(x)$ in terms of $u$, at which point I am stuck. My current attempt was to reduce the power down from 8 to 2, and express it as $(1-u^2)$, but that gives me a wrong answer (this assignment checks for us before submission).

Any help is appreciated.

Thanks.

Answer 1

If $$\cos^2 x = 1-\sin^2 x$$ then $$\cos^8 x = (1-\sin^2 x)^4.$$

Answer 2

Using $u=\sin x$, you have $du=\cos x\,dx$, and also $\cos^2x=1-u^2$, so $\cos^8x=(1-u^2)^4$. Thus your integral becomes $$ \int u^7(1-u^2)^4\,du $$