Solving reduction formula problem

Question

I wish you are doing well. I would like to get and answer for this question: $\int\sin(x)^2dx=\frac{x}{2}-\frac{\sin{2x}}{4}+c$

So here we need to use reduction formula to prove that this expression is valid.

Answer 1

HINT:

Use $\cos2x=1-2\sin^2x\iff\sin^2x=?$

Answer 2

By reduction formula:

$$\sin^2x = \frac{1}{2} - \frac{\cos(2x)}{2}$$

Hence your integral is the integral of those two terms, that is:

$$\frac{1}{2}x - \frac{1}{2}\left(\frac{\sin(2x)}{2}\right) = \frac{1}{2}x - \frac{1}{4}\sin(2x)$$

Plus a constant $C$.

Answer 3

Remember the identity $\cos{2x}=1-2\sin^2x$, so $\sin^2x=\frac{1}{2}-\frac{\cos{2x}}{2}$. Hence

$$\int \sin^2x dx=\int\left(\frac{1}{2}-\frac{\cos{2x}}{2}\right)dx$$

I think you can integrate it by yourself.