What is the integral of e^x(2+3e^x)^5 ?

The integral of e^x(2+3e^x)^5 is 1/18 (2+3e^x)^6+C

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integral of e^x(2+3e^x)^5

\int e^{x} (2 + 3 e^{x})^{5} d x

\frac{1}{18} (2 + 3 e^{x})^{6} + C

Solution steps

\int e^{x} (2 + 3 e^{x})^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{3} d u

= \int \frac{u ^{5}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{3} \cdot \frac{u ^{6}}{6}

Substitute back

u = 2 + 3 e^{x}

= \frac{1}{3} \cdot \frac{( 2 + 3 e ^{x} ) ^{6}}{6}

Simplify

\frac{1}{3} \cdot \frac{( 2 + 3 e ^{x} ) ^{6}}{6} : \frac{1}{18} (2 + 3 e^{x})^{6}

= \frac{1}{18} (2 + 3 e^{x})^{6}

Add a constant to the solution

= \frac{1}{18} (2 + 3 e^{x})^{6} + C

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What is the integral of e^x(2+3e^x)^5 ?
The integral of e^x(2+3e^x)^5 is 1/18 (2+3e^x)^6+C