query: Show that $\int_{0}^{\pi}{\sin^5(x)\over 1+\cos^3(x)}\mathrm dx =\ln{3}?$
positive_passages.docid: 26608
positive_passages.text: How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$
negative_passages.docid: 71988
negative_passages.text: Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$
negative_passages.docid: 25797
negative_passages.text: Integral of $ x \ln( \sin (x))$ from 0 to $ \pi $
negative_passages.docid: 99753
negative_passages.text: How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
negative_passages.docid: 109385
negative_passages.text: Evaluating the integral $\int_{-\ln 2}^{\ln 2} e^{-x}(\sin x+x)^{1/3}dx$
negative_passages.docid: 9653
negative_passages.text: Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$.
negative_passages.docid: 144039
negative_passages.text: Computing $ \int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx$
negative_passages.docid: 124209
negative_passages.text: How is $\ln(-1) = i\pi$?
negative_passages.docid: 154065
negative_passages.text: $\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums?
negative_passages.docid: 100485
negative_passages.text: How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?
negative_passages.docid: 100869
negative_passages.text: How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
negative_passages.docid: 83703
negative_passages.text: Show that $\int_{0}^{\infty}|\frac{\sin x}{x}|\,dx=\infty $
negative_passages.docid: 31716
negative_passages.text: Calculate :$\int_{0}^{2\pi }e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$
negative_passages.docid: 148613
negative_passages.text: Integrate $\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$
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: 69000
negative_passages.text: Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
negative_passages.docid: 28771
negative_passages.text: Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $