query: Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $
positive_passages.docid: 16775
positive_passages.text: Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
negative_passages.docid: 12859
negative_passages.text: Find $\int \frac{x^3}{x^2-1} \:\mathrm{d}x$
negative_passages.docid: 106812
negative_passages.text: Find $\int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx$
negative_passages.docid: 151649
negative_passages.text: Exact Values of the integal $\int_0^\infty \frac{r^{n-1}}{(1+r^2)^{\frac{s}{2}}}\,dr$
negative_passages.docid: 119209
negative_passages.text: Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
negative_passages.docid: 126046
negative_passages.text: solving the integral $ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $
negative_passages.docid: 24558
negative_passages.text: Show that $\frac{1}{\sqrt{2\pi}} \cdot \left(\int_{- \infty}^{\infty} e^{itx} \cdot e^{-\frac{1}{2}t^2 } dx \right) = e^{-\frac{1}{2}t^2}$
negative_passages.docid: 96486
negative_passages.text: Double Integral $\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2$
negative_passages.docid: 60654
negative_passages.text: Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$
negative_passages.docid: 84765
negative_passages.text: Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$
negative_passages.docid: 15625
negative_passages.text: Computing $\int_0^\infty\mathrm{d} x\frac{x}{e^x+1}$ with contour integration
negative_passages.docid: 29425
negative_passages.text: $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral
negative_passages.docid: 102932
negative_passages.text: Integral $\int_0^\infty \frac{\arctan(x) dx}{x(1+x^2)}$?
negative_passages.docid: 31964
negative_passages.text: Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
negative_passages.docid: 128572
negative_passages.text: How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?
negative_passages.docid: 147108
negative_passages.text: Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$
negative_passages.docid: 13747
negative_passages.text: Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$