What is the integral of xe^{x-2} ?

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

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integral of xe^{x-2}

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

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

Solution steps

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

Apply Integration By Parts

: \frac{1}{2} e^{x - 2} x^{2} - \int \frac{1}{2} e^{x - 2} x^{2} d x

= \frac{1}{2} e^{x - 2} x^{2} - \int \frac{1}{2} e^{x - 2} x^{2} d x

\int \frac{1}{2} e^{x - 2} x^{2} d x = \frac{1}{2} (e^{x - 2} x^{2} - 2 e^{x - 2} x + 2 e^{x - 2})

= \frac{1}{2} e^{x - 2} x^{2} - \frac{1}{2} (e^{x - 2} x^{2} - 2 e^{x - 2} x + 2 e^{x - 2})

Simplify

\frac{1}{2} e^{x - 2} x^{2} - \frac{1}{2} (e^{x - 2} x^{2} - 2 e^{x - 2} x + 2 e^{x - 2}) : e^{x - 2} x - e^{x - 2}

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

Add a constant to the solution

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

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\int \frac{1}{3} e^{2 - x} d x

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What is the integral of xe^{x-2} ?
The integral of xe^{x-2} is e^{x-2}x-e^{x-2}+C