Reverse Chain Rule Intergration

Question

The question states; determine the following using reverse chain rule;

$$\int\sin x\cos^5x\, dx$$

Can you show me how to do this when you let $U =$ either $\sin$ or $\cos$?

Answer 1

Let $u = \cos x \implies du = -\sin x dx$

Then you have $$\int \sin x \cos^5 x \,dx = \;-\int (\underbrace{\cos x}_{\large u})^5 \underbrace{(-\sin x dx)}_{\large du} = -\int u^5 \,du$$

Can you take it from here?