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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

u = 1 + e^{2 x}

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

Simplify

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

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

Add a constant to the solution

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

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