Gaussian integral
\[\int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}\]
Dirichlet integral
\[\int_0^{\infty} \frac{\sin x}{x}\, dx = \frac{\pi}{2}\]
Basel-adjacent
\[\int_0^1 \frac{\ln x}{x-1}\, dx = \frac{\pi^2}{6}\]
Fresnel-style
\[\int_0^\infty \sin(x^2)\, dx = \sqrt{\frac{\pi}{8}}\]
Beta function
\[\int_0^1 x^{a-1}(1-x)^{b-1}\, dx = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\]
Ahmed's integral
\[\int_0^1 \frac{\tan^{-1}\!\sqrt{x^2+2}}{(x^2+1)\sqrt{x^2+2}}\, dx = \frac{5\pi^2}{96}\]