New posts in improper-integrals

How can I study the convergence of the improper integral $\int_{0}^{ \infty} \frac{\sin(x)}{x+1} \, \mathrm dx\,$?

real-analysis integration convergence-divergence improper-integrals

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

calculus integration definite-integrals improper-integrals closed-form

Show $\sigma_{X}^{2}(t)=\begin{cases} x_{0}\frac{\beta}{\alpha}e^{\alpha t}[e^{\alpha t}-1], & \alpha \neq 0\\ x_{0}\beta t, & \alpha = 0 \end{cases}$

real-analysis probability integration improper-integrals stochastic-calculus

Prove that $\int\limits_{-\infty}^{\infty} \frac{e^{-x}}{1+e^{-2\pi x}}\,dx=\frac1{2\sin\left(\frac{1}{2}\right)}$

integration complex-analysis improper-integrals residue-calculus complex-integration

Proof of $\int_0^\infty\frac{\left (1- e^{\pi\sqrt3x}\cos(\pi x )\right )e^{-2\pi x/\sqrt3}}{x(1+x^3)(1+x^3/2^3)(1+x^3/3^3)\dots}~dx=0.$

integration improper-integrals indefinite-integrals infinite-product

Solution to the "near-Gaussian" integral $\int_{0}^{\infty} e^{- \Lambda \sqrt{(z^2+a)^2+b^2}}\mathrm{d}z$

integration definite-integrals improper-integrals

An alternative way to define improper integrals

real-analysis integration improper-integrals

Convergence of $\int_0^\infty\frac{\sin x}{x}dx$

calculus integration improper-integrals

A probabilistic integral $\int_{-\infty}^{\infty}e^{-x^2/2\sigma^2}\arcsin\left(1-2\left|\lfloor x\rceil-x\right|\right)\,dx$

integration definite-integrals improper-integrals

Ramanujan's Master Theorem relation to Analytic Continuation

integration complex-analysis power-series improper-integrals

Challenging integral $\int_{0}^{1}\frac{x\operatorname{li}(x)}{x^2+1}dx$

definite-integrals logarithms power-series improper-integrals

Taking the derivative under a principal value integral

complex-analysis improper-integrals

Ramanujan Integral Identity assuming $\alpha\beta=\pi^2$

integration improper-integrals

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

calculus real-analysis integration definite-integrals improper-integrals

Integral $ \int_{0}^{\infty} \ln x\left[\ln \left( \frac{x+1}{2} \right) - \frac{1}{x+1} - \psi \left( \frac{x+1}{2} \right) \right] \mathrm{d}x $

real-analysis definite-integrals improper-integrals special-functions

Convergence/Divergence of $\int_e^\infty \frac{\sin x}{x \ln x}\;dx$

sequences-and-series convergence-divergence improper-integrals

Convergence or Divergence using Limits

calculus improper-integrals

How to demonstrate the equality of these integral representations of $\pi$?

calculus integration definite-integrals improper-integrals pi

Evaluation and generalisation of $\int_0^\infty\int_0^\infty\sin y\frac{\operatorname{gd}(xy)}{\cosh(xy)}\mathrm dx\mathrm dy=\frac{\pi^3}{16}$

integration trigonometry improper-integrals closed-form multiple-integral

A $\log$ integral with a parameter

calculus sequences-and-series integration improper-integrals gamma-function