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

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

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

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

- 2 1 + e^{- x} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= - 2 u^{\frac{1}{2}}

Substitute back

u = 1 + e^{- x}

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

Apply radical rule:

= - 2 1 + e^{- x}

Add a constant to the solution

= - 2 1 + e^{- x} + C

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