What is the integral of (e^{-3x})/(5-2e^{-3x)} ?

The integral of (e^{-3x})/(5-2e^{-3x)} is 1/6 ln|5-2e^{-3x}|+C

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integral of (e^{-3x})/(5-2e^{-3x)}

\int \frac{e ^{- 3 x}}{5 - 2 e ^{- 3 x}} d x

\frac{1}{6} ln 5 - 2 e^{- 3 x} + C

Solution steps

\int \frac{e ^{- 3 x}}{5 - 2 e ^{- 3 x}} d x

Apply u-substitution

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = 5 - 2 e^{- 3 x}

= \frac{1}{6} ln 5 - 2 e^{- 3 x}

Add a constant to the solution

= \frac{1}{6} ln 5 - 2 e^{- 3 x} + C

What is the integral of (e^{-3x})/(5-2e^{-3x)} ?
The integral of (e^{-3x})/(5-2e^{-3x)} is 1/6 ln|5-2e^{-3x}|+C