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

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

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integral of (e^{-2x})/(e^{-2x)+1}

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

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

Solution steps

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

Apply u-substitution

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

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}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = e^{- 2 x} + 1

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

Add a constant to the solution

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

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