New posts in definite-integrals

Harmonic number

definite-integrals

the value of $\lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$

calculus real-analysis limits definite-integrals

Remarkable integral: $\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) \mathrm dx= -\frac{13 \pi ^2}{72}$?

integration definite-integrals

Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$

calculus real-analysis integration complex-analysis definite-integrals

Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$

calculus integration trigonometry definite-integrals closed-form

Find $\int_{0}^{\infty }\frac{\cos x-\cos x^2}{x}\mathrm dx$

integration definite-integrals improper-integrals

Closed form expression for the harmonic sum $\sum\limits_{n=1}^{\infty}\frac{H_{2n}}{n^2\cdot4^n}{2n \choose n}$

real-analysis sequences-and-series definite-integrals binomial-coefficients harmonic-numbers

Find the value of : $\lim_{n\to\infty}[(n+1)\int_{0}^{1}x^{n}\ln(1+x)dx]$.

integration definite-integrals

$\int_0^1\frac{\ln x\ln^2(1-x^2)}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\zeta(3)-2\pi\ln^32$

integration sequences-and-series definite-integrals logarithms riemann-zeta

Integral $\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}$

calculus real-analysis integration complex-analysis definite-integrals

Suggestion for Computing an Integral

calculus integration analysis definite-integrals

Evaluation of a dilogarithmic integral

sequences-and-series definite-integrals closed-form harmonic-numbers polylogarithm

For any given function $f\colon [0,1]\to\Bbb R$, what is $\int_0^1\frac{f(x)}{f(x)+f(1-x)}dx$?

calculus real-analysis integration definite-integrals closed-form

Is there a rapider or more elegant way to evaluate $\int_0^{+\infty} \frac{\cos(\pi x)\ \text{d}x}{e^{2\pi \sqrt{x}}-1}$?

calculus integration definite-integrals special-functions residue-calculus

Evaluate $\int_0^1\frac {\{2x^{-1}\}}{1+x}\,\mathrm d x$

definite-integrals

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}\text dx$

calculus real-analysis integration definite-integrals improper-integrals

Calculating $\int_0^1 \frac{\operatorname{arctanh}\left(\sqrt{1-\frac{u}{2}}\right)\sqrt{\frac{2 \pi \sqrt{1-u}}{u-2}+\pi } }{u\sqrt{1-u}} \, du$

calculus real-analysis integration definite-integrals

A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

calculus integration definite-integrals hypergeometric-function polylogarithm

Generalized logarithm integral $\int^1_0 \frac{\log(1+x)}{1+a^2x^2}\, dx$

calculus integration definite-integrals

Find $\int_0^1\frac{\ln^2(1-x)}{x}\ dx$

calculus integration sequences-and-series definite-integrals