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

The integral of (e^x)/((1+e^x)) is ln|1+e^x|+C

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

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

ln ∣ 1 + e^{x} ∣ + C

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 1 + e^{x}

= ln ∣ 1 + e^{x} ∣

Add a constant to the solution

= ln ∣ 1 + e^{x} ∣ + C

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