What is the integral of 1/2 e^xx^2 ?

The integral of 1/2 e^xx^2 is 1/2 e^xx^2-e^xx+e^x+C

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integral of 1/2 e^xx^2

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

\frac{1}{2} e^{x} x^{2} - e^{x} x + e^{x} + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 1/2 e^xx^2 ?
The integral of 1/2 e^xx^2 is 1/2 e^xx^2-e^xx+e^x+C