Definite Integral from 0 to Half Pi of x over Sine x

Theorem

 * $\displaystyle \int_0^{\pi/2} \frac x {\sin x} \rd x = 2 G$

where $G$ is Catalan's constant.

Proof
From Definite Integral from $0$ to $1$ of $\dfrac {\arctan x} x$, we have:


 * $\displaystyle \int_0^1 \frac {\arctan x} x \rd x = G$

Let:


 * $x = \tan \theta$

By Derivative of Tangent Function, we have:


 * $\displaystyle \frac {\d x} {\d \theta} = \sec^2 \theta$

We have, by Arctangent of Zero is Zero:


 * as $x \to 0$, $\theta \to 0$.

We also have, by Arctangent of One:


 * as $x \to 1$, $\theta \to \dfrac \pi 4$

We therefore have:

giving:


 * $\displaystyle \int_0^{\pi/2} \frac \phi {\sin \phi} \rd \phi = 2 G$