What is the integral of (2e^x+4)/(2e^x-4) ?

The integral of (2e^x+4)/(2e^x-4) is -x+2ln|e^x-2|+C

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integral of (2e^x+4)/(2e^x-4)

\int \frac{2 e ^{x} + 4}{2 e ^{x} - 4} d x

- x + 2 ln ∣ e^{x} - 2 ∣ + C

Solution steps

\int \frac{2 e ^{x} + 4}{2 e ^{x} - 4} d x

Simplify

\frac{2 e ^{x} + 4}{2 e ^{x} - 4} : \frac{e ^{x} + 2}{e ^{x} - 2}

= \int \frac{e ^{x} + 2}{e ^{x} - 2} d x

Expand

\frac{e ^{x} + 2}{e ^{x} - 2} : \frac{e ^{x}}{e ^{x} - 2} + \frac{2}{e ^{x} - 2}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{e ^{x}}{e ^{x} - 2} d x + \int \frac{2}{e ^{x} - 2} d x

\int \frac{e ^{x}}{e ^{x} - 2} d x = ln ∣ e^{x} - 2 ∣

\int \frac{2}{e ^{x} - 2} d x = - x + ln ∣ e^{x} - 2 ∣

= ln ∣ e^{x} - 2 ∣ - x + ln ∣ e^{x} - 2 ∣

Simplify

= - x + 2 ln ∣ e^{x} - 2 ∣

Add a constant to the solution

= - x + 2 ln ∣ e^{x} - 2 ∣ + C

What is the integral of (2e^x+4)/(2e^x-4) ?
The integral of (2e^x+4)/(2e^x-4) is -x+2ln|e^x-2|+C