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

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

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integral of (3e^x)/(e^x-2)

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

3 ln ∣ e^{x} - 2 ∣ + C

Solution steps

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

= 3 ln ∣ e^{x} - 2 ∣

Add a constant to the solution

= 3 ln ∣ e^{x} - 2 ∣ + C

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