query: Showing $\int\limits_0^{\pi}\log(\sin(\theta))d\theta = -\pi \log(2)$
positive_passages.docid: 828
positive_passages.text: Computing the integral of $\log(\sin x)$
negative_passages.docid: 26608
negative_passages.text: How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$
negative_passages.docid: 135653
negative_passages.text: An integral$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$ which I can calculate but can't understand it.
negative_passages.docid: 12604
negative_passages.text: Evaluate $\int_0^\pi xf(\sin x)dx$
negative_passages.docid: 10036
negative_passages.text: Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
negative_passages.docid: 149866
negative_passages.text: Evaluating $\int_0^{2\pi} \frac {\cos(\theta)}{5-3\cos(\theta)} \, d\theta$ using residue theorem
negative_passages.docid: 31063
negative_passages.text: Evaluating $\int_0^2(\tan^{-1}(\pi x)-\tan^{-1} x)\,\mathrm{d}x$
negative_passages.docid: 8358
negative_passages.text: Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
negative_passages.docid: 80903
negative_passages.text: Help with integrating $\int_0^{\infty} \frac{(\log x)^2}{x^2 + 1} \operatorname d\!x$ - contour integration?
negative_passages.docid: 25797
negative_passages.text: Integral of $ x \ln( \sin (x))$ from 0 to $ \pi $
negative_passages.docid: 93610
negative_passages.text: What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?
negative_passages.docid: 149913
negative_passages.text: Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$
negative_passages.docid: 25489
negative_passages.text: Evaluate the Integral: $\int e^{2\theta}\ \sin 3\theta\ d\theta$
negative_passages.docid: 130967
negative_passages.text: How to show that $\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi?$
negative_passages.docid: 137060
negative_passages.text: Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$
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: 147759
negative_passages.text: Integral $ \dfrac { \int_0^{\pi/2} (\sin x)^{\sqrt 2 + 1} dx} { \int_0^{\pi/2} (\sin x)^{\sqrt 2 - 1} dx} $