Integrating $\int_0^{\pi/2} \sin x \cdot \cos^2 x \, dx$ by substitution

Question

How to find Integration.

$$\int _{ 0 }^{ \pi /2 }{ \sin x\cdot\cos^ 2 x } \,dx$$

My work: Let $\cos x =u$.

Upper Limit = $0$

Lower Limit = $1$

$$\int _{ 1 }^{ 0 }{ \sin(x)\cdot u^ 2 } \,\frac{du}{-\sin x}$$

Can anyone help after this?

Answer 1

Hint: with $$t=\cos(x)$$ we get $$dt=-\sin(x)dx$$ For $x=0$ we get $$t=\cos(0)=1$$ For $x=\frac{\pi}{2}$ we get $$t=\cos(\frac{\pi}{2})=0$$