What is the integral of e^x(2+3e^x)^2 ?

The integral of e^x(2+3e^x)^2 is 4e^x+6e^{2x}+3e^{3x}+C

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integral of e^x(2+3e^x)^2

\int e^{x} (2 + 3 e^{x})^{2} d x

4 e^{x} + 6 e^{2 x} + 3 e^{3 x} + C

Solution steps

\int e^{x} (2 + 3 e^{x})^{2} d x

Expand

e^{x} (2 + 3 e^{x})^{2} : 4 e^{x} + 12 e^{2 x} + 9 e^{3 x}

= \int 4 e^{x} + 12 e^{2 x} + 9 e^{3 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 4 e^{x} d x + \int 12 e^{2 x} d x + \int 9 e^{3 x} d x

\int 4 e^{x} d x = 4 e^{x}

\int 12 e^{2 x} d x = 6 e^{2 x}

\int 9 e^{3 x} d x = 3 e^{3 x}

= 4 e^{x} + 6 e^{2 x} + 3 e^{3 x}

Add a constant to the solution

= 4 e^{x} + 6 e^{2 x} + 3 e^{3 x} + C

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What is the integral of e^x(2+3e^x)^2 ?
The integral of e^x(2+3e^x)^2 is 4e^x+6e^{2x}+3e^{3x}+C