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

The integral of e^x(1+4e^x)^2 is 1/12 (1+4e^x)^3+C

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

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

\frac{1}{12} (1 + 4 e^{x})^{3} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{1}{4} \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + 4 e^{x}

= \frac{1}{4} \cdot \frac{( 1 + 4 e ^{x} ) ^{3}}{3}

Simplify

\frac{1}{4} \cdot \frac{( 1 + 4 e ^{x} ) ^{3}}{3} : \frac{1}{12} (1 + 4 e^{x})^{3}

= \frac{1}{12} (1 + 4 e^{x})^{3}

Add a constant to the solution

= \frac{1}{12} (1 + 4 e^{x})^{3} + C

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What is the integral of e^x(1+4e^x)^2 ?
The integral of e^x(1+4e^x)^2 is 1/12 (1+4e^x)^3+C