What is the integral of e^x(1/2 x^2+x) ?

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

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integral of e^x(1/2 x^2+x)

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

Add a constant to the solution

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

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