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

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

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

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

arctan (e^{x}) + C

Solution steps

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

Apply u-substitution

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

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

Use the common integral:

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

= arctan (u)

Substitute back

u = e^{x}

= arctan (e^{x})

Add a constant to the solution

= arctan (e^{x}) + C

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