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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u ^{3}} d u

= \int \frac{1}{2 u ^{3}} 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 \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= \frac{1}{2} (- \frac{1}{2 u ^{2}})

Substitute back

u = 1 + e^{2 x}

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

Simplify

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

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

Add a constant to the solution

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

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