calculus - Integral $int_0^{infty} frac{ln cos^2 x}{x^2}dx=-pi$

$$I:=\int_0^{\infty} \frac{\ln \cos^2 x}{x^2}dx=-\pi.$$ Using $2\cos^2 x=1+\cos 2x$ failed me because I ran into two divergent integrals after using $\ln(ab)=\ln a + \ln b$ since I obtained $\int_0^\infty x^{-2}dx$ and $\int_0^\infty (1+\cos^2 x)dx$ which both diverge. Perhaps we should try a complex analysis approach? I also tried writing $$I(\alpha)=\int_0^\infty \frac{\ln \cos^2 \alpha \,x}{x^2}dx$$ and obtained $$-\frac{dI(\alpha)}{d\alpha}=2\int_0^\infty \frac{\tan \alpha x}{x}dx=\int_{-\infty}^\infty\frac{\tan \alpha x}{x}dx.$$ Taking a second derivative $$I''(\alpha)=\int_{-\infty}^\infty {\sec^2 (\alpha x)}\, dx$$ Random Variable pointed out how to continue from the integral after the 1st derivative, but is it possible to work with this integral $\sec^2 \alpha x$? Thanks

Answer

Let the desired integral be denoted by $I$. Note that

$$\eqalign{ 2I&=\int_{-\infty}^\infty\frac{\ln(\cos^2x)}{x^2}dx= \sum_{n=-\infty}^{+\infty}\left(\int_{n\pi}^{(n+1)\pi}\frac{\ln(\cos^2x)}{x^2}dx\right)\cr &=\sum_{n=-\infty}^{+\infty}\left(\int_{0}^{\pi}\frac{\ln(\cos^2x)}{(x+n\pi)^2}dx\right) \cr &=\int_{0}^{\pi}\left(\sum_{n=-\infty}^{+\infty} \frac{1}{(x+n\pi)^2}\right)\ln(\cos^2x)dx \cr &=\int_{0}^{\pi}\frac{\ln(\cos^2x)}{\sin^2x}dx \cr }$$ where the interchange of the signs of integration and summation is justified by the fact that the integrands are all negative, and we used the well-known expansion: $$\sum_{n=-\infty}^{+\infty} \frac{1}{(x+n\pi)^2}=\frac{1}{\sin^2x}.\tag{1}$$ Now using the symmetry of the integrand arround the line $x=\pi/2$, we conclude that $$\eqalign{ I&=\int_{0}^{\pi/2}\frac{\ln(\cos^2x)}{\sin^2x}dx\cr &=\Big[-\cot(x)\ln(\cos^2x)\Big]_{0}^{\pi/2}+\int_0^{\pi/2}\cot(x)\frac{-2\cos x\sin x}{\cos^2x}dx\cr &=0-2\int_0^{\pi/2}dx=-\pi. }$$ and the announced conclusion follows.$\qquad\square$

Remark: Here is a proof of $(1)$ that does not use residue theorem. Consider $\alpha\in(0,1)$, and let $f_\alpha$ be the $2\pi$-periodic function that coincides with $x\mapsto e^{i\alpha x}$ on the interval $(-\pi,\pi)$. It is easy to check that the exponential Fourier coefficients of $f_\alpha$ are given by

$$C_n(f_\alpha)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f_\alpha(x)e^{-inx}dx=\sin(\alpha\pi)\frac{(-1)^n}{\alpha \pi-n\pi}$$ So, by Parseval's formula we have $$\sum_{n\in\Bbb{Z}}\vert C_n(f_\alpha)\vert^2=\frac{1}{2\pi}\int_{-\pi}^\pi\vert f_\alpha(x)\vert^2dx$$ That is $$\sin^2(\pi\alpha) \sum_{n\in\Bbb{Z}}\frac{1}{(\alpha\pi-n\pi)^2}=1$$ and we get $(1)$ by setting $x=\alpha\pi\in(0,\pi)$.