What is the integral of (2e^{3x})/(1-e^{3x)} ?

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

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integral of (2e^{3x})/(1-e^{3x)}

\int \frac{2 e ^{3 x}}{1 - e ^{3 x}} d x

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

Solution steps

\int \frac{2 e ^{3 x}}{1 - e ^{3 x}} d x

Take the constant out:

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

= 2 \cdot \int \frac{e ^{3 x}}{1 - e ^{3 x}} d x

Apply u-substitution

: \int - \frac{1}{3 u} d u

= 2 \cdot \int - \frac{1}{3 u} d u

Take the constant out:

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

= 2 (- \frac{1}{3} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{3} ln ∣ u ∣)

Substitute back

u = 1 - e^{3 x}

= 2 (- \frac{1}{3} ln 1 - e^{3 x})

Simplify

2 (- \frac{1}{3} ln 1 - e^{3 x}) : - \frac{2}{3} ln 1 - e^{3 x}

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

Add a constant to the solution

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

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