An Example

Consider the integral

$$\int_{-\pi}^{\pi} \sin^2 nx dx = ?$$ (6)

$$= \int_{-\pi}^{\pi} \left (\frac{e^{\imath n x}-e{-\imath n x}}{2\imath}\right )^2 dx$$

$$= -\frac{1}{4}\int_{-\pi}^{\pi} \left (e^{2\imath n x}+e{-2\imath n x} - 2\right )^2 dx$$

$$= -\frac{1}{4}\int_{-\pi}^{\pi} \left( 2 \cos 2 n x - 2\right )^2 dx$$

$$= -\frac{1}{4} \left .\left( 2 \frac{\sin 2 n x}{-2n} - 2x\right )\right \vert _{-\pi}^{\pi} dx$$

$$-\frac{2}{4} ( - \pi - (-\pi)) = \pi$$