The Catalan Constant

Let's take $x= \sin(u)$ , the integral becomes $\displaystyle\int_0^{\frac{π}{2}} u^2 \cot(u)\cosec(u) du$

Using Integrating by parts ,the integral becomes

$\displaystyle-[u^2 \cosec(u) ]_0^{π/2} +2\int_0^{\frac{π}{2}} u \cosec(u) du = -\dfrac{π^2}{4} + \int_0^{\frac{π}{2}} u \cosec(u) du$

Using Parts again $\displaystyle -\dfrac{π^2}{4} -[2u\log(\cot(u/2)) ] _0^{\frac{π}{2}} +2\int_0^{\frac{π}{2}} \log(\cot(\frac{u}{2}) )du$

Now $\displaystyle\int_0^{\frac{π}{2}} \log(\cot(\frac{u}{2})) du =^{\textrm{Substituting u=2t}} = -2\int_0^{\frac{π}{4}} \log(\tan(t))dt$

Substituting $\tan(t)=y$

$\displaystyle=-2\int_0^1 \dfrac{\log(y)}{y^2+1} dy =[-2\log(y) \arctan(y)]_0^1 +2\int_0^1 \dfrac{\tan^{-1}(y)}{y} dy$

$\displaystyle 2 \sum_{n=0}^∞ (-1)^{n} \dfrac{1}{2n+1} \int_0^1 x^{2n} dx=2 \sum_{n=0}^∞ (-1)^n \dfrac{1}{(2n+1)^2} = 2G$

Final answer will be $\boxed{\displaystyle 4G-\dfrac{π^2}{4}}$

Visual representation of the integral: