Newbetuts

for $\int_0^{\pi/2} \frac{\cos^\alpha(x)}{x\sqrt{\sin^\beta(x)}} \,dx$ to converge, $\alpha,\beta=?$

An integral which could be split into even and odd functions

How to prove $\int_{0}^{\pi}\frac{\sin^n{x}}{(1+r^2-2r\cos{x})^{(n+2)/2}}\,\mathrm dx=\frac{1}{1-r^2}\int_{0}^{\pi}\sin^n{x}\,\mathrm dx$

Can a substitution cause a convergent definite integral to diverge?

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

On the integral $\int_{(0,1)^n}\frac{\prod\sin\theta_k}{\sum\sin\theta_k}d\mu$

When is $\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$?

Prove that $\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for any $a>0$

Evaluating $\int\limits_0^\infty{\frac{1}{1+x^2+x^\alpha}dx}$

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

Integral boundaries of random variable when $|x|+|y| \leq 1$

Integral with arctan and e: $\int_{0}^{\infty}\frac{\arctan(x^{3})}{e^{2\pi x}-1}\,\mathrm dx$

Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$

Evaluating $\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$ [duplicate]

Find $\int_{0}^{\frac{\pi}{2}} \ln(\sin(x)) \ln( \cos(x))\,\mathrm dx$

Alternative ways to evaluate $\int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$

New posts in definite-integrals

Proof of this integration shortcut: $\int_a^b \frac{dx}{\sqrt{(x-a)(b-x)}}=\pi$

Integral that evaluates to 42 [closed]

Trouble understanding Blagouchine's extensions to the Malmsten integral

Calculate $\int_0^{2\pi} \tan \frac{\theta}8 d\theta $ using complex analysis

for $\int_0^{\pi/2} \frac{\cos^\alpha(x)}{x\sqrt{\sin^\beta(x)}} \,dx$ to converge, $\alpha,\beta=?$

An integral which could be split into even and odd functions

How to prove $\int_{0}^{\pi}\frac{\sin^n{x}}{(1+r^2-2r\cos{x})^{(n+2)/2}}\,\mathrm dx=\frac{1}{1-r^2}\int_{0}^{\pi}\sin^n{x}\,\mathrm dx$

Can a substitution cause a convergent definite integral to diverge?

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

On the integral $\int_{(0,1)^n}\frac{\prod\sin\theta_k}{\sum\sin\theta_k}d\mu$

When is $\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$?

Prove that $\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for any $a>0$

Evaluating $\int\limits_0^\infty{\frac{1}{1+x^2+x^\alpha}dx}$

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

Integral boundaries of random variable when $|x|+|y| \leq 1$

Integral with arctan and e: $\int_{0}^{\infty}\frac{\arctan(x^{3})}{e^{2\pi x}-1}\,\mathrm dx$

Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$

Evaluating $\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$ [duplicate]

Find $\int_{0}^{\frac{\pi}{2}} \ln(\sin(x)) \ln( \cos(x))\,\mathrm dx$

Alternative ways to evaluate $\int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$