What is the integral of (3e^{-x})/(4-e^{-x)} ?

The integral of (3e^{-x})/(4-e^{-x)} is 3ln|4-e^{-x}|+C

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integral of (3e^{-x})/(4-e^{-x)}

\int \frac{3 e ^{- x}}{4 - e ^{- x}} d x

3 ln 4 - e^{- x} + C

Solution steps

\int \frac{3 e ^{- x}}{4 - e ^{- x}} d x

Take the constant out:

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

= 3 \cdot \int \frac{e ^{- x}}{4 - e ^{- x}} d x

Apply u-substitution

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

= 3 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 ln ∣ u ∣

Substitute back

u = 4 - e^{- x}

= 3 ln 4 - e^{- x}

Add a constant to the solution

= 3 ln 4 - e^{- x} + C

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What is the integral of (3e^{-x})/(4-e^{-x)} ?
The integral of (3e^{-x})/(4-e^{-x)} is 3ln|4-e^{-x}|+C