What is the integral of (e^{x+1})/(e^x-5) ?

The integral of (e^{x+1})/(e^x-5) is eln|e^x-5|+C

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integral of (e^{x+1})/(e^x-5)

\int \frac{e ^{x + 1}}{e ^{x} - 5} d x

e ln ∣ e^{x} - 5 ∣ + C

Solution steps

\int \frac{e ^{x + 1}}{e ^{x} - 5} d x

Apply u-substitution

: \int \frac{e}{u} d u

= \int \frac{e}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= e ln ∣ u ∣

Substitute back

u = e^{x} - 5

= e ln ∣ e^{x} - 5 ∣

Add a constant to the solution

= e ln ∣ e^{x} - 5 ∣ + C

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What is the integral of (e^{x+1})/(e^x-5) ?
The integral of (e^{x+1})/(e^x-5) is eln|e^x-5|+C