Evaluate the indefinite integral of xe^-x

by RoRi

November 11, 2015

Compute the following indefinite integral,

$\int xe^{-x} \, dx.$

To evaluate this integral we use integration by parts, letting

$\begin{align*} u &= x & du &= dx, \\ dv &= e^{-x} \, dx & v &= -e^{-x}. \end{align*}$

Therefore,

$\begin{align*} \int xe^{-x} \, dx &= -xe^{-x} + \int e^{-x} \, dx \\ &= -xe^{-x} - e^{-x} + C \\ &= -e^{-x} (x + 1) + C. \end{align*}$

Stumbling Robot