calculus - Some integral with sine

$$\begin{align} & \int_{0}^{+\infty }{\frac{\sin px}{1+{{\text{e}}^{qx}}}}\text{d}x ,\ \ p,\ q>0\\ \\ \\ & \int_{0}^{+\infty }{{{\left( \frac{\sin x}{x} \right)}^{n}}\text{d}x} \\ \end{align}$$

Answer

I will sketch an outline of the first integral. There are convergence questions, and a sum that evaluates into a special function for which we will need a derivation.

Write the integral as

$$\int_0^{\infty} dx \: \frac{\sin{p x} e^{-q x}}{1+e^{-q x}}$$

Taylor expand the denominator:

$$\int_0^{\infty} dx \: \sin{p x} \, e^{-q x} \sum_{n=0}^{\infty} (-1)^n e^{-n q x}$$

Reverse the order of sum and integral. Again, the justification is not trivial (see, e.g., Abel's Theorem):

$$\begin{align} \sum_{n=0}^{\infty} (-1)^n \int_0^{\infty} dx \: \sin{p x} \, e^{-(n+1) q x} &= \Im[\sum_{n=0}^{\infty} (-1)^n \int_0^{\infty} dx \: e^{-[(n+1) q -i p]x} \\ &= \Im{\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)q - i p}} \\ &= p \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)^2 q^2 + p^2} \\ &= \frac{1}{2 p} \left [ 1 - \frac{p^2}{q^2} \sum_{n=-\infty}^{\infty} \frac{(-1)^n}{n^2 + \frac{p^2}{q^2}} \right ] \end{align}$$

I will not provide a derivation of the following closed form yet (perhaps in an update):

$$\begin{align} \sum_{n=-\infty}^{\infty} \frac{(-1)^n}{n^2 + a^2} = \frac{\pi}{a} \mathrm{csch}{\pi a} & (a>0) \\ \end{align}$$

(I can say that one derivation uses residues in the complex plane.) Using this result, we find that

$$\int_0^{\infty} dx \: \frac{\sin{p x}}{1+e^{q x}} = \frac{1}{2 p} - \frac{\pi}{2 q} \mathrm{csch}{\pi \frac{p}{q}}$$