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Integration by parts of xe^x

The product $xe^x$ is integrated by parts. You differentiate $x$ , which becomes 1, and integrate $e^x$ , which stays itself. The new integral is then only the exponential function, and the answer is $(x - 1)e^x$ plus a constant. If you differentiate the answer, you should get $xe^x$ back. In a definite integral you insert the limits into the whole expression.

Try it yourself

The integral as area

\int xe^x\,dx = (x - 1)e^x + C

integration by parts

Symbols

$x$	the factor that is differentiated
$e^x$	the factor that is integrated

Example

$\int_0^1 xe^x\,dx = [(x - 1)e^x]_0^1 = 0 - (-1) = 1$ . From $1$ to $2$ the value is $(2 - 1)e^2 - (1 - 1)e = e^2 \approx 7.389$ .

The check is to differentiate

(x - 1)e^x

. The product rule gives

e^x + (x - 1)e^x = xe^x

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Part of Mathematics R2: Integration techniques.