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

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

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integral of (e^{2x})/(e^{4x)+1}

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 ( u ^{2} + 1 )} d u

= \int \frac{1}{2 ( u ^{2} + 1 )} 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} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

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

Substitute back

u = e^{2 x}

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

Add a constant to the solution

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

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