What is the integral of xe^xx ?

The integral of xe^xx is e^xx^2-2e^xx+2e^x+C

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integral of xe^xx

\int x e^{x} x d x

e^{x} x^{2} - 2 e^{x} x + 2 e^{x} + C

Solution steps

\int x e^{x} x d x

Apply Integration By Parts

: x (e^{x} x - e^{x}) - \int e^{x} x - e^{x} d x

= x (e^{x} x - e^{x}) - \int e^{x} x - e^{x} d x

\int e^{x} x - e^{x} d x = e^{x} x - 2 e^{x}

= x (e^{x} x - e^{x}) - (e^{x} x - 2 e^{x})

Simplify

x (e^{x} x - e^{x}) - (e^{x} x - 2 e^{x}) : e^{x} x^{2} - 2 e^{x} x + 2 e^{x}

= e^{x} x^{2} - 2 e^{x} x + 2 e^{x}

Add a constant to the solution

= e^{x} x^{2} - 2 e^{x} x + 2 e^{x} + C

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