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

The integral of-(e^x)/(2+e^x) is -ln|2+e^x|+C

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

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

- ln ∣ 2 + e^{x} ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = 2 + e^{x}

= - ln ∣ 2 + e^{x} ∣

Add a constant to the solution

= - ln ∣ 2 + e^{x} ∣ + C

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What is the integral of-(e^x)/(2+e^x) ?
The integral of-(e^x)/(2+e^x) is -ln|2+e^x|+C