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

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

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integral of e^x(e^x+x)

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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