What is the integral of (xe^x)^2 ?

The integral of (xe^x)^2 is 1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x})+C

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integral of (xe^x)^2

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

\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + C

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

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

\int e^{2 x} x d x = \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

= \frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

Add a constant to the solution

= \frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + C

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What is the integral of (xe^x)^2 ?
The integral of (xe^x)^2 is 1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x})+C