What is the integral of e^x(1+2e^x)^4 ?

The integral of e^x(1+2e^x)^4 is 1/10 (1+2e^x)^5+C

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integral of e^x(1+2e^x)^4

\int e^{x} (1 + 2 e^{x})^{4} d x

\frac{1}{10} (1 + 2 e^{x})^{5} + C

Solution steps

\int e^{x} (1 + 2 e^{x})^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{2} d u

= \int \frac{u ^{4}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{1}{2} \cdot \frac{u ^{5}}{5}

Substitute back

u = 1 + 2 e^{x}

= \frac{1}{2} \cdot \frac{( 1 + 2 e ^{x} ) ^{5}}{5}

Simplify

\frac{1}{2} \cdot \frac{( 1 + 2 e ^{x} ) ^{5}}{5} : \frac{1}{10} (1 + 2 e^{x})^{5}

= \frac{1}{10} (1 + 2 e^{x})^{5}

Add a constant to the solution

= \frac{1}{10} (1 + 2 e^{x})^{5} + C

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What is the integral of e^x(1+2e^x)^4 ?
The integral of e^x(1+2e^x)^4 is 1/10 (1+2e^x)^5+C