New posts in definite-integrals

Log integrals I

integration definite-integrals logarithms polylogarithm

Finding $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\} \left\{\frac{y}{z}\right\}\left\{\frac{z}{x}\right\} dx\space dy\space dz $

calculus integration definite-integrals

Showing $\int_0^{\infty } \frac{1}{e^{2 x}+2 e^x \cos (x)+1} \, dx=\frac{\log (2)}{2}-\pi \sum _{n=0}^{\infty } \frac{1}{e^{\pi (2 n+1)}+1}$

integration sequences-and-series definite-integrals

On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$?

integration sequences-and-series definite-integrals binomial-coefficients closed-form

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

definite-integrals improper-integrals

How does one convert an integrand of the form $\frac{x\sinh x-t\sinh t}{\sinh^2x-\sinh^2t}$ into the form $\frac{\ln(x^2-t^2+1)}{\sinh^2x-\sinh^2t}$?

definite-integrals logarithms contour-integration gamma-function hyperbolic-functions

Does $ \int _1^{\infty }\frac{\sinh (a \log (x))}{\sqrt{x}} $ converge or diverge?

integration definite-integrals

Evaluate $\int_{0}^{\frac {\pi}{3}}x\log(2\sin\frac {x}{2})\,dx$

calculus integration definite-integrals special-functions improper-integrals

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

integration sequences-and-series definite-integrals summation closed-form

Evaluate integral: $\int_0^{\frac{\pi}{2}}\ln(a^2\cos^2 x +b^2\sin^2x)dx$?

definite-integrals summation

Is there a known closed form solution to $\int_0^1\frac{\ln(1+x^{2n})}{1+x^2} \,dx$?

definite-integrals closed-form

Ways to prove $ \int_0^1 \frac{\ln^2(1+x)}{x}dx = \frac{\zeta(3)}{4}$?

integration sequences-and-series definite-integrals polylogarithm

Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

complex-analysis definite-integrals contour-integration

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

integration definite-integrals closed-form

Integral$\int_0^\infty \ln x\,\exp(-\frac{1+x^4}{2\alpha x^2}) \frac{x^4+\alpha x^2- 1}{x^4}dx$?

calculus real-analysis integration complex-analysis definite-integrals

The negative integral meaning

calculus integration definite-integrals indefinite-integrals

A surprising dilogarithm integral identity arising from a generalised point enclosure problem

geometry definite-integrals geometric-probability polylogarithm

Evaluating a slow sum

calculus sequences-and-series definite-integrals

Evaluate $\int_0^1\frac{x^3 - x^2}{\ln x}\,\mathrm dx$? [duplicate]

calculus integration definite-integrals improper-integrals

How to integrate $\int\limits_{0}^{\pi/2}\frac{dx}{\cos^3{x}+\sin^3{x}}$?

calculus integration definite-integrals closed-form