What is the integral of (e^x)/(5+e^{2x)} ?

The integral of (e^x)/(5+e^{2x)} is 1/(sqrt(5))arctan((e^x)/(sqrt(5)))+C

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integral of (e^x)/(5+e^{2x)}

\int \frac{e ^{x}}{5 + e ^{2 x}} d x

\frac{1}{5} arctan (\frac{e ^{x}}{5}) + C

Solution steps

\int \frac{e ^{x}}{5 + e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{5 + u ^{2}} d u

= \int \frac{1}{5 + u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{5 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= \frac{1}{5} arctan (v)

Substitute back

= \frac{1}{5} arctan (\frac{e ^{x}}{5})

Add a constant to the solution

= \frac{1}{5} arctan (\frac{e ^{x}}{5}) + C

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What is the integral of (e^x)/(5+e^{2x)} ?
The integral of (e^x)/(5+e^{2x)} is 1/(sqrt(5))arctan((e^x)/(sqrt(5)))+C