What is the integral of e^x(2+3e^x)^4 ?

The integral of e^x(2+3e^x)^4 is 16e^x+48e^{2x}+72e^{3x}+54e^{4x}+81/5 e^{5x}+C

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integral of e^x(2+3e^x)^4

\int e^{x} (2 + 3 e^{x})^{4} d x

16 e^{x} + 48 e^{2 x} + 72 e^{3 x} + 54 e^{4 x} + \frac{81}{5} e^{5 x} + C

Solution steps

\int e^{x} (2 + 3 e^{x})^{4} d x

Expand

e^{x} (2 + 3 e^{x})^{4} : 16 e^{x} + 96 e^{2 x} + 216 e^{3 x} + 216 e^{4 x} + 81 e^{5 x}

= \int 16 e^{x} + 96 e^{2 x} + 216 e^{3 x} + 216 e^{4 x} + 81 e^{5 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 16 e^{x} d x + \int 96 e^{2 x} d x + \int 216 e^{3 x} d x + \int 216 e^{4 x} d x + \int 81 e^{5 x} d x

\int 16 e^{x} d x = 16 e^{x}

\int 96 e^{2 x} d x = 48 e^{2 x}

\int 216 e^{3 x} d x = 72 e^{3 x}

\int 216 e^{4 x} d x = 54 e^{4 x}

\int 81 e^{5 x} d x = \frac{81}{5} e^{5 x}

= 16 e^{x} + 48 e^{2 x} + 72 e^{3 x} + 54 e^{4 x} + \frac{81}{5} e^{5 x}

Add a constant to the solution

= 16 e^{x} + 48 e^{2 x} + 72 e^{3 x} + 54 e^{4 x} + \frac{81}{5} e^{5 x} + C

What is the integral of e^x(2+3e^x)^4 ?
The integral of e^x(2+3e^x)^4 is 16e^x+48e^{2x}+72e^{3x}+54e^{4x}+81/5 e^{5x}+C