# Tagged Questions

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### Choose appropriate contour for a complex integral

I have a problem to solve integral $$I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}}$$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
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### Integral $\displaystyle \int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that ...
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### Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$ [duplicate]

Hi I'm trying to show that $$I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}.$$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
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### Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta.$$ The ...
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### Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
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### LogSine Generating Fn $\int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$\int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta.$$ ...
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Hello I am trying to integrate $$I=\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta$$ which is similar to Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ... 2answers 184 views ### Integral$\int_0^{\pi/2}dx\ln \sinh x$$$I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2.$$ I am trying to calculate these integrals. We know the similar looking integrals $$\int_0^{\pi/2}dx\ln ... 1answer 125 views ### Integral \int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx Hey I am trying to integrate$$ I_n:=\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$Thanks. This integral is old. I am also looking for literature on these integrals ... 1answer 58 views ### Integration of trigonometric functions times a simple rational function using residues In the course of my research I have found a few integrals that I would like to have closed-form answers to:$$\int_{c- i \infty}^{c+ i \infty} \frac{1}{z-1} \frac{8 \pi^4 \cot{ \big( \frac{\pi}{6} z ... 1answer 81 views ### Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$Hi I am trying to integrate $$\int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0.$$ These integrals can be ... 1answer 42 views ### Integral$\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$$$I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n.$$ Evaluate I for$n \to \infty$and z real. We can consider$z\geq 0$due to the symmetry of$\mathcal{F}$given by $$... 0answers 99 views ### Integral \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx Hey I am trying to integrate$$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$This integral is old. I am also looking for literature on these ... 1answer 286 views ### Integrate \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx Hi I am trying to come up with a closed form expression for$$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$I am interested in this general case in terms of p. For small p, we can ... 0answers 72 views ### Integrate \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi EDIT/UPDATE: I DO NOT NEED A SOLUTION. SEE SOS440 COMMENT FOR A FULL DETAILED SOLUTION. Hi I am trying to integrate$$ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi, ... 2answers 131 views ### Integral$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $I am trying to solve this integral $$\int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$ A closed form does exist despite the looks of the integrand. ... 3answers 115 views ### Integrate$\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx$I need to solve for the parameter$\alpha$after I calculate the integral.$$\mathcal{R}(\alpha,\beta)=\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx, \ \ \beta >0$$ The result of this ... 1answer 93 views ### Integral$I=\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx$Hello can you please help me solve this integral $$\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx=-\frac{2\pi}{3}(\pi^2+24\ln 2).$$ I am trying to work through all logarithmic integrals. Note, ... 4answers 255 views ### Integrate$ \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$I am trying to evaluate this integral. $$I=\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$$ Note $$\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, \ |x| < 1.$$ I was trying to do ... 3answers 341 views ### Integral$I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$Hi I am stuck on showing that $$\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8}$$ where G is the Catalan constant and$\zeta(3)$is the Riemann zeta function. Explictly ... 2answers 119 views ### Integral$I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $$$I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0.$$ I am trying to solve this integral. This is from the old high school ... 2answers 128 views ### Integrating$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $Compute $$\int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx.$$ I am not sure how to start this one...I am thinking of a substitution to get started. 2answers 236 views ### Integral$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$Hello there I am trying to calculate $$\int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$$ NOT using mathematica, matlab, etc. We are given that$\sigma, \omega$are complex. Note, the ... 1answer 101 views ### Integral$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$Hi I'm trying to solve this integral Fourier Transform $$\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k))$$ where sgn(k)$=1$for k>1 and$-1$for k<1. I am ... 2answers 148 views ### Integral$\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$Hi I am trying to calculate, $$\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$$ where$a,b$are positive real constants. I Know$\ln(xy)=\ln x +\ln y$, but I do not ... 2answers 162 views ### Integral$ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral Calculate $$I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1.$$ I am trying to solve this integral$I_1$. I know how to solve a related integral$I_2$$$I_2:=\int_0^1 \frac{\ln \ln ... 0answers 165 views ### Integral =\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx. I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is$$ ... 4answers 117 views ### Computing$\int_0^\infty\mathrm{d} x\frac{x}{e^x+1}$with contour integration Let's set: $$\int_0^\infty\mathrm{d}x\frac{x}{e^x+1}=I.$$ I would like to compute it using, presumably, the methods of complex analysis and contour integration. 2answers 379 views ### Integral$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$. I am trying to calculate $$I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}.$$ Note, we can expand the log in the integral to ... 1answer 33 views ### Is there no analytic form of$\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$? I am trying to find an analytic answer for$\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$but it doesn't seem to be in any of the integral tables that I've looked in. I don't think ... 2answers 185 views ### Computing the integral$ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $Integrate $$\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi.$$ Something that may help$(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ... 3answers 218 views ### Differentiation wrt parameter$\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx\$

Use differentiation with respect to parameter obtaining a differential equation to solve $$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx$$ No complex variables, only this approach. Interesting ...
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### definiteinteggral

The integral is given by $$\int_0^1 \frac{\ln (1-x)\ln x}{1+x} dx = \frac{1}{8}\big(-\pi^2\ln(4) +13\zeta(3)\big).$$ Any ideas how to prove? We cannot solve the integral so easily because we cannot ...
Trying to show $$\int_0^1 \frac{\ln x \cdot \ln(1+x)}{1-x}dx=-\frac{1}{4}\pi^2 \ln(2)+\zeta(3).$$ I am unsure how to approach this integral as I do not know how to use a power series representation ...
How can we prove $$\int_0^1 \frac{\ln x \cdot \ln(1+x)}{1+x}dx=-\frac{\zeta(3)}{8}?$$ This has been one of the integrals that came out of an integral from another post on here, but no solution to ...