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

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

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integral of xe^{x+2}

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

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

Solution steps

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

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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