query: Evaluate $\int_0^\infty\frac{\sin^4x}{x^2}dx$ with $\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2}$
positive_passages.docid: 64480
positive_passages.text: How do I find $\int_{0}^{\infty} \frac{\sin^4 x}{x^2}\,dx$?
negative_passages.docid: 18672
negative_passages.text: Integrate $\int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$
negative_passages.docid: 3068
negative_passages.text: Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
negative_passages.docid: 1416
negative_passages.text: closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$
negative_passages.docid: 11129
negative_passages.text: Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
negative_passages.docid: 16775
negative_passages.text: Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
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: 88372
negative_passages.text: Integrate : $\int \frac{\sin x}{\sin4x} \,dx$
negative_passages.docid: 38511
negative_passages.text: Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$
negative_passages.docid: 95933
negative_passages.text: How to evaluate the integral $\int_0^{\pi}\frac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx$?
negative_passages.docid: 141892
negative_passages.text: How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$
negative_passages.docid: 143734
negative_passages.text: Evaluating $\int \frac{\sin\left(x\right)}{1+x^2}dx$
negative_passages.docid: 21375
negative_passages.text: The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$
negative_passages.docid: 137549
negative_passages.text: Given $\int_0^{\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}$, evaluate $\int_0^{\infty}e^{-a^2x^2-\frac{b^2}{x^2}}dx $
negative_passages.docid: 132000
negative_passages.text: Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths
negative_passages.docid: 77834
negative_passages.text: Is it possible to compute $\int_{-\infty}^\infty {x \sin x\over x^4+1}dx$ without using complex analysis?
negative_passages.docid: 93174
negative_passages.text: Evaluate integral $\int_0^\pi \sin^4\left(x+\sin 3x\right)dx$