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There are some functions whose antiderivatives ''cannot'' be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
:<math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math> (see also [[Gamma function]])
:<math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (the [[Gaussian integral]])
:<math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math> (see also [[Bernoulli number]])
:<math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math>
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:<math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}</math>
:<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (where <math>\Gamma(z)</math> is the [[Gamma function]])
:<math>\int_{-\infty}^\infty \exp\left[-(ax^2+bx+c)\right]\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math> (where <math> \exp\left[
:<math>\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (where <math>I_{0}(x)</math> is the modified [[Bessel function]] of the first kind)
:<math>\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \sqrt{x^2 + y^2} </math>
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