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

The integral of e^x(3+2e^x)^4 is 81e^x+108e^{2x}+72e^{3x}+24e^{4x}+16/5 e^{5x}+C

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

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

81 e^{x} + 108 e^{2 x} + 72 e^{3 x} + 24 e^{4 x} + \frac{16}{5} e^{5 x} + C

Solution steps

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

Expand

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

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

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

\int 216 e^{2 x} d x = 108 e^{2 x}

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

\int 96 e^{4 x} d x = 24 e^{4 x}

\int 16 e^{5 x} d x = \frac{16}{5} e^{5 x}

= 81 e^{x} + 108 e^{2 x} + 72 e^{3 x} + 24 e^{4 x} + \frac{16}{5} e^{5 x}

Add a constant to the solution

= 81 e^{x} + 108 e^{2 x} + 72 e^{3 x} + 24 e^{4 x} + \frac{16}{5} e^{5 x} + C

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What is the integral of e^x(3+2e^x)^4 ?
The integral of e^x(3+2e^x)^4 is 81e^x+108e^{2x}+72e^{3x}+24e^{4x}+16/5 e^{5x}+C