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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{6 u} d u

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

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

Add a constant to the solution

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

ln (sin (x))

y^{^{'}} = y x^{2}

t^{3} cos (3 t)

\frac{d}{d x} (1 + (x)^{2})

\int x 9^{x} d x

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