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

The integral of (e^{-x})/(e^{-x)+e^x} is -1/2 ln|e^{-2x}+1|+C

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integral of (e^{-x})/(e^{-x)+e^x}

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

- \frac{1}{2} ln e^{- 2 x} + 1 + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 v} d v

= - \int \frac{1}{2 v} d v

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}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{2} ln ∣ v ∣

Substitute back

= - \frac{1}{2} ln e^{2 (- x)} + 1

Simplify

= - \frac{1}{2} ln e^{- 2 x} + 1

Add a constant to the solution

= - \frac{1}{2} ln e^{- 2 x} + 1 + C

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What is the integral of (e^{-x})/(e^{-x)+e^x} ?
The integral of (e^{-x})/(e^{-x)+e^x} is -1/2 ln|e^{-2x}+1|+C