I've been given this integral, and I have no idea how to evaluate it. It has a really nice answer of $\frac{1}{2}$ but I have no idea. I tried converting it into a series. I tried differentiating under the integral. I can't find a way of using complex numbers. The only thing I can think of is complex analysis which I don't know. Is there a way to solve it without complex analysis? If not, I wouldn't mind that as a solution.
Here's the integral:
$$\int_0^\infty \frac{1}{(x+1)(\pi^2+\ln(x)^2)}dx$$
Best Answer
$$\int_{0}^{+\infty}\frac{dx}{(x+1)(\pi^2+\log^2 x)}=\int_{0}^{1}\frac{dx}{(x+1)(\pi^2+\log^2 x)}+\int_{0}^{1}\frac{dx}{x(x+1)(\pi^2+\log^2 x)}$$ equals $$ \int_{0}^{1}\frac{dx}{x(\pi^2+\log^2 x)}\stackrel{x\mapsto e^t}{=}\int_{-\infty}^{0}\frac{dt}{\pi^2+t^2}=\int_{0}^{+\infty}\frac{du}{\pi^2+u^2}=\left[\frac{\arctan(u/\pi)}{\pi}\right]_{0}^{+\infty}=\frac{1}{2}.$$
An overkill is to exploit the integral representation of Gregory coefficients.