What is the integral of 1/3 xe^x ?

The integral of 1/3 xe^x is 1/3 (e^xx-e^x)+C

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integral of 1/3 xe^x

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

\frac{1}{3} (e^{x} x - e^{x}) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{3} \cdot \int x e^{x} d x

Apply Integration By Parts

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

= \frac{1}{3} (e^{x} x - \int e^{x} d x)

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

= \frac{1}{3} (e^{x} x - e^{x})

Add a constant to the solution

= \frac{1}{3} (e^{x} x - e^{x}) + C

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