What is the integral of e^{2x}(1+e^{2x})^3 ?

The integral of e^{2x}(1+e^{2x})^3 is 1/8 (1+e^{2x})^4+C

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integral of e^{2x}(1+e^{2x})^3

\int e^{2 x} (1 + e^{2 x})^{3} d x

\frac{1}{8} (1 + e^{2 x})^{4} + C

Solution steps

\int e^{2 x} (1 + e^{2 x})^{3} d x

Apply u-substitution

: \int \frac{u ^{3}}{2} d u

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

Apply the Power Rule

: \frac{u ^{4}}{4}

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

Substitute back

u = 1 + e^{2 x}

= \frac{1}{2} \cdot \frac{( 1 + e ^{2 x} ) ^{4}}{4}

Simplify

\frac{1}{2} \cdot \frac{( 1 + e ^{2 x} ) ^{4}}{4} : \frac{1}{8} (1 + e^{2 x})^{4}

= \frac{1}{8} (1 + e^{2 x})^{4}

Add a constant to the solution

= \frac{1}{8} (1 + e^{2 x})^{4} + C

\frac{d}{d w} (2 l + 2 w)

\int (t^{2} + 1) d t

\frac{\partial}{\partial x} (2 x^{3} + 2 x^{2} y - 3 y^{3})

\int_{- 10}^{10} 100 - x^{2} d x

\frac{d y}{d x} = x^{2} x - 2

What is the integral of e^{2x}(1+e^{2x})^3 ?
The integral of e^{2x}(1+e^{2x})^3 is 1/8 (1+e^{2x})^4+C