$\int\sin^2(Cx)\,dx$ from a manual - need proof

Question

In the book of quantum mechanics I came across an integral which was supposed to be from a manual ($C$ is a constant):

\begin{align} \int\limits_{0}^d \sin^2\left( C x \right)\, d x = \left.\left(\frac{x}{2}- \frac{\sin(2Cx)}{4C}\right)\right|_0^d \end{align}

Where can I read more about this? I would be glad if anyone could provide me a proof.

Answer 1

$$\sin ^2 Cx=\dfrac{1-\cos 2Cx}{2}$$ $$\int \sin ^2 Cx\;dx=\int\dfrac{1-\cos 2Cx}{2}\;dx$$ $$\int\dfrac 12-\dfrac{\cos 2Cx}{2}dx$$ $$\dfrac {x}{2}-\dfrac{\sin 2Cx}{2\cdot 2C}$$ $$\dfrac {x}{2}-\dfrac{\sin 2Cx}{4C}$$ Now just put given limits $$\left[\dfrac {x}{2}-\dfrac{\sin 2Cx}{4C}\right]_0^d$$ $$\dfrac {d}{2}-\dfrac{\sin 2Cd}{4C}$$

Answer 2

Hints:

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

$$\int\sin^2xdx=\frac12\int(1-\cos2x)dx=\frac12\left(x-\frac12\sin2x\right)$$

Finally, do a simple substitution

$$u=Cx\implies\;dx=\frac1Cdu\;,\;\;x=0\implies u=0\;,\;\;x=d\implies u=Cd\ldots\ldots$$