What is the integral of (e^{2x}+10)^3 ?

The integral of (e^{2x}+10)^3 is 1/6 e^{6x}+15/2 e^{4x}+150e^{2x}+1000x+C

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integral of (e^{2x}+10)^3

\int (e^{2 x} + 10)^{3} d x

\frac{1}{6} e^{6 x} + \frac{15}{2} e^{4 x} + 150 e^{2 x} + 1000 x + C

Solution steps

\int (e^{2 x} + 10)^{3} d x

Expand

(e^{2 x} + 10)^{3} : e^{6 x} + 30 e^{4 x} + 300 e^{2 x} + 1000

= \int e^{6 x} + 30 e^{4 x} + 300 e^{2 x} + 1000 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^{6 x} d x + \int 30 e^{4 x} d x + \int 300 e^{2 x} d x + \int 1000 d x

\int e^{6 x} d x = \frac{1}{6} e^{6 x}

\int 30 e^{4 x} d x = \frac{15}{2} e^{4 x}

\int 300 e^{2 x} d x = 150 e^{2 x}

\int 1000 d x = 1000 x

= \frac{1}{6} e^{6 x} + \frac{15}{2} e^{4 x} + 150 e^{2 x} + 1000 x

Add a constant to the solution

= \frac{1}{6} e^{6 x} + \frac{15}{2} e^{4 x} + 150 e^{2 x} + 1000 x + C

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What is the integral of (e^{2x}+10)^3 ?
The integral of (e^{2x}+10)^3 is 1/6 e^{6x}+15/2 e^{4x}+150e^{2x}+1000x+C