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Definite integrals

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of these functions over some common intervals can be calculated. A few useful definite integrals are given below.

\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi

\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi

\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}

\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}

\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}

\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)

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