New posts in definite-integrals

Evaluate $\int_0^1 \log \left( \frac{x^2+\sqrt{3}x+1}{x^2-\sqrt{3}x+1} \right) \frac{dx}{x} $

definite-integrals

Prove that $ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$

calculus real-analysis integration inequality definite-integrals

Prove that $\int_0^1|f''(x)|dx\ge4.$

calculus real-analysis inequality definite-integrals functional-inequalities

Prove that $\int\limits_0^1 x^a(1-x)^{-1}\ln x \,dx = -\sum\limits_{n=1}^\infty \frac{1}{(n+a)^2}$

calculus real-analysis integration sequences-and-series definite-integrals

Integration of some floor functions

calculus integration definite-integrals indefinite-integrals ceiling-and-floor-functions

Evaluating $\int_0^{\infty} \frac {e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$

integration definite-integrals improper-integrals

How to solve this definite integral? It cannot be solved using simple integration by parts.

calculus integration definite-integrals

Evaluate $\int_0^{\pi/4} \frac {\sin x} {x \cos^2 x} \mathrm d x$

definite-integrals

Does this integral have a closed form?

calculus integration definite-integrals

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh \pi x +\pi x\sinh \pi x}{\cosh^2 \pi x}\frac{dx}{x^2}=4-\pi$?

calculus real-analysis integration complex-analysis definite-integrals

Evaluating this integral using the Gamma function

calculus integration definite-integrals improper-integrals gamma-function

Proving that $\int_0^\infty\sin(x)dx=1$

calculus integration limits definite-integrals fake-proofs

Proving $\int_0^\infty\left(\frac{x^xe^{-x}}{\Gamma(x+1)}-\frac1{\sqrt{2\pi x}}\right)dx=-\frac13$

integration sequences-and-series definite-integrals gamma-function

Solving $\int_0^{\infty} \frac{\sin^m(x)}{x^n} dx$ for $m, n \in \mathbb{Z}^+$

real-analysis calculus integration definite-integrals

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

complex-analysis definite-integrals contour-integration residue-calculus

Definite Integral $\int_0^1\frac{\ln(x^2-x+1)}{x^2-x}\,\mathrm{d}x$

definite-integrals

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

calculus integration multivariable-calculus definite-integrals closed-form

Evaluate $\int_{0}^{\pi/2}{\frac{d\theta}{{\left(\cos^3{\theta}+\sin^3{\theta}\right)}^{2/3}}}$

integration definite-integrals trigonometric-integrals

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

integration definite-integrals closed-form

Calculating ${\int_{-\infty}^{\infty} \frac{\cos(\omega x)}{x^{2} + 25}\,{\rm d}x}$ using contour integration

complex-analysis definite-integrals contour-integration