New posts in improper-integrals

Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$

improper-integrals pi golden-ratio euler-mascheroni-constant

Integral $\int_{-\infty}^\infty \frac{\exp{(-x^2)}}{1+x^4}dx$

calculus integration complex-analysis improper-integrals

Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$

calculus integration special-functions improper-integrals gamma-function

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

calculus definite-integrals improper-integrals closed-form polylogarithm

Mathematica gives: $\int_{0}^{\infty}{\cos(x^n)-\cos(x^{2n})\over x}\cdot{\ln{x}}\mathrm dx={12\gamma^2-\pi^2\over 2(4n)^2}$

calculus integration definite-integrals improper-integrals euler-mascheroni-constant

Uniform convergence of a parametric integral. Which solution is correct?

improper-integrals uniform-convergence

An almost Fresnel integral

integration improper-integrals

Show that $K=K'$ if and only if $k=(\sqrt{2}-1)^2$ (Ahlfors)

complex-analysis integration improper-integrals elliptic-integrals

Prove that $\int_0^\infty\,\frac{\sin(kx)}{x(x^2+1)}\,\text{d}x=\frac{\pi}{2}\,\left(1-\exp(-k)\right)$ for all $k\in\mathbb{R}_{\ge0}$.

real-analysis integration definite-integrals improper-integrals contour-integration

Show $\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$ for $t \gt 0$

calculus integration definite-integrals improper-integrals

Proving that $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}$ by induction

calculus integration improper-integrals

Does $\int\limits_0^\infty \cos(x^3 -x) \, \mathrm dx$ converge?

improper-integrals

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

calculus integration definite-integrals improper-integrals closed-form

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

calculus real-analysis integration definite-integrals improper-integrals

Solving $\int_0^{\infty} \sin(ax^2)\sin(b/x^2)\,\mathrm dx$

calculus definite-integrals improper-integrals

Calculating $\int_0^\infty \frac{\sin(x)}{x} \frac{\sin(x / 3)}{x / 3} \frac{\sin(x / 5)}{x / 5} \cdots \frac{\sin(x / 15)}{x / 15} \ dx$

integration improper-integrals closed-form products trigonometric-integrals

Improper Integral $\int_0^\infty\left(\frac{\tanh(x)}{x^3}-\frac1{x^2\cosh^2(x)}\right)dx = \frac{7\zeta(3)}{\pi^2} $

calculus integration definite-integrals improper-integrals closed-form

Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$

calculus integration definite-integrals improper-integrals closed-form

Generalized Riemann Integral: Improper Version

integration functional-analysis improper-integrals banach-spaces

Generalization of Improper Integral

real-analysis calculus integration improper-integrals riemann-integration