What is the integral of xe^xy ?

The integral of xe^xy is y(e^xx-e^x)+C

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integral of xe^xy

\int x e^{x} y d x

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

Solution steps

\int x e^{x} y d x

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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