query: How does one show that $\int_0^\infty \left|\frac{\sin x} x\right| \, dx=\infty$?
positive_passages.docid: 143857
positive_passages.text: How to prove absolute summability of sinc function?
negative_passages.docid: 105707
negative_passages.text: A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$
negative_passages.docid: 19514
negative_passages.text: $\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show?
negative_passages.docid: 13747
negative_passages.text: Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
negative_passages.docid: 68821
negative_passages.text: Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
negative_passages.docid: 3704
negative_passages.text: Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
negative_passages.docid: 107296
negative_passages.text: How do I prove $\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}{\cos(a-b)}$?
negative_passages.docid: 123899
negative_passages.text: $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$
negative_passages.docid: 13287
negative_passages.text: Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$
negative_passages.docid: 87242
negative_passages.text: Ways to prove $ \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$
negative_passages.docid: 31792
negative_passages.text: show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $
negative_passages.docid: 101267
negative_passages.text: Show: $\int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt$
negative_passages.docid: 8829
negative_passages.text: How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
negative_passages.docid: 158279
negative_passages.text: $\int_0^{\pi/2}\ln(\sin(x))$?
negative_passages.docid: 40537
negative_passages.text: How to prove $\int_0^\infty e^{-x^2}cos(2bx) dx = \frac{\sqrt{\pi}}{2} e^{-b^2}$
negative_passages.docid: 100642
negative_passages.text: Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$
negative_passages.docid: 10547
negative_passages.text: $\int_0^\infty \frac{\sin(x^2+ax)}{x}\,dx$ converges