What is the integral of e^{5x}(e^{5x}+2)^7 ?

The integral of e^{5x}(e^{5x}+2)^7 is 1/40 (e^{5x}+2)^8+C

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integral of e^{5x}(e^{5x}+2)^7

\int e^{5 x} (e^{5 x} + 2)^{7} d x

\frac{1}{40} (e^{5 x} + 2)^{8} + C

Solution steps

\int e^{5 x} (e^{5 x} + 2)^{7} d x

Apply u-substitution

: \int \frac{u ^{7}}{5} d u

= \int \frac{u ^{7}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{1}{5} \cdot \frac{u ^{8}}{8}

Substitute back

u = e^{5 x} + 2

= \frac{1}{5} \cdot \frac{( e ^{5 x} + 2 ) ^{8}}{8}

Simplify

\frac{1}{5} \cdot \frac{( e ^{5 x} + 2 ) ^{8}}{8} : \frac{1}{40} (e^{5 x} + 2)^{8}

= \frac{1}{40} (e^{5 x} + 2)^{8}

Add a constant to the solution

= \frac{1}{40} (e^{5 x} + 2)^{8} + C

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What is the integral of e^{5x}(e^{5x}+2)^7 ?
The integral of e^{5x}(e^{5x}+2)^7 is 1/40 (e^{5x}+2)^8+C