Evaluate the integral of x times e to the power of x dx.

C. e^x(x - 1) + C — Apply integration by parts where u equals x and dv equals e to the power of x dx. The formula integral u dv equals uv minus integral v du gives x e to the power of x minus integral e to the power of x dx.

JEE Main · MathematicsMedium

$\int x e^{x} d x$

A. $e^{x} + C$
B. $x e^{x} + C$
C. $e^{x} (x - 1) + C$
D. $e^{x} (x + 1) + C$

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Correct answer: C — $e^{x} (x - 1) + C$

Solution

Using integration by parts: \int u d v = uv - \int v d u . Let u = x, d v = e^{x} d x . Then \int x e^{x} d x = x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + C = e^{x} (x - 1) + C .

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∫xexdx

Solution

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$\int x e^{x} d x$