Integral of $$$x^{p - 1} e^{- x}$$$ with respect to $$$x$$$

The calculator will find the integral/antiderivative of $$$x^{p - 1} e^{- x}$$$ with respect to $$$x$$$, with steps shown.

Find $$$\int x^{p - 1} e^{- x}\, dx$$$.

This integral (Incomplete Gamma Function) does not have a closed form:

$${\color{red}{\int{x^{p - 1} e^{- x} d x}}} = {\color{red}{\left(- \Gamma\left(p, x\right)\right)}}$$

Therefore,

$$\int{x^{p - 1} e^{- x} d x} = - \Gamma\left(p, x\right)$$

Add the constant of integration:

$$\int{x^{p - 1} e^{- x} d x} = - \Gamma\left(p, x\right)+C$$

$$$\int x^{p - 1} e^{- x}\, dx = - \Gamma\left(p, x\right) + C$$$A

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