What is the integral of (e^{3x})/(e^{3x)-8} ?

The integral of (e^{3x})/(e^{3x)-8} is 1/3 ln|e^{3x}-8|+C

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integral of (e^{3x})/(e^{3x)-8}

\int \frac{e ^{3 x}}{e ^{3 x} - 8} d x

\frac{1}{3} ln e^{3 x} - 8 + C

Solution steps

\int \frac{e ^{3 x}}{e ^{3 x} - 8} d x

Apply u-substitution

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = e^{3 x} - 8

= \frac{1}{3} ln e^{3 x} - 8

Add a constant to the solution

= \frac{1}{3} ln e^{3 x} - 8 + C

What is the integral of (e^{3x})/(e^{3x)-8} ?
The integral of (e^{3x})/(e^{3x)-8} is 1/3 ln|e^{3x}-8|+C