New posts in closed-form

Arithmetic-quadratic mean and other "means by limits of means"

real-analysis closed-form means

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

Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$

calculus closed-form infinite-product hyperbolic-functions

Closed form for the sum of even fibonacci numbers?

summation fibonacci-numbers closed-form project-euler

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

A limit related to super-root (tetration inverse).

calculus limits convergence-divergence closed-form tetration

Integral $\int_0^{\frac{\pi}{2}} x^2 \sqrt{\sin x} \, dx$

integration closed-form

Proving $\int_0^1 \frac{(\ln(x))^5}{1+x} \mathrm{d}x = -\frac{31\pi^6}{252}$

calculus integration definite-integrals contour-integration closed-form

Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

integration definite-integrals special-functions closed-form bessel-functions

Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?

combinatorics summation generating-functions closed-form

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

real-analysis sequences-and-series trigonometry summation closed-form

The integral $\ J(m,n):=\int_0^1 \frac{x^m}{x^n+1}dx$

calculus integration closed-form

Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ [duplicate]

calculus integration sequences-and-series limits closed-form

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

calculus definite-integrals improper-integrals contour-integration closed-form

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

calculus integration definite-integrals closed-form

Value of this convergent series: $\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$

sequences-and-series convergence-divergence closed-form

Closed form of finite Euler sum $\sum_{k=1}^n \frac{ H_{k}}{(2k+1)}$

summation closed-form harmonic-numbers

Prove $\int_{\frac{\pi}{20}}^{\frac{3\pi}{20}} \ln \tan x\,\,dx= - \frac{2G}{5}$

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

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

integration definite-integrals physics closed-form