What is the integral of (1-5e^x)^2e^x ?

The integral of (1-5e^x)^2e^x is e^x-5e^{2x}+25/3 e^{3x}+C

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integral of (1-5e^x)^2e^x

\int (1 - 5 e^{x})^{2} e^{x} d x

e^{x} - 5 e^{2 x} + \frac{25}{3} e^{3 x} + C

Solution steps

\int (1 - 5 e^{x})^{2} e^{x} d x

Expand

(1 - 5 e^{x})^{2} e^{x} : e^{x} - 10 e^{2 x} + 25 e^{3 x}

= \int e^{x} - 10 e^{2 x} + 25 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 e^{x} d x - \int 10 e^{2 x} d x + \int 25 e^{3 x} d x

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

\int 10 e^{2 x} d x = 5 e^{2 x}

\int 25 e^{3 x} d x = \frac{25}{3} e^{3 x}

= e^{x} - 5 e^{2 x} + \frac{25}{3} e^{3 x}

Add a constant to the solution

= e^{x} - 5 e^{2 x} + \frac{25}{3} e^{3 x} + C

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What is the integral of (1-5e^x)^2e^x ?
The integral of (1-5e^x)^2e^x is e^x-5e^{2x}+25/3 e^{3x}+C