What is the integral of ((e^x)/(e^x+5)) ?

The integral of ((e^x)/(e^x+5)) is ln|e^x+5|+C

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integral of ((e^x)/(e^x+5))

\int (\frac{e ^{x}}{e ^{x} + 5}) d x

ln ∣ e^{x} + 5 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = e^{x} + 5

= ln ∣ e^{x} + 5 ∣

Add a constant to the solution

= ln ∣ e^{x} + 5 ∣ + C

\int (9 x + 2) (3 x - 5) d x

\int (e^{5 x} - \frac{5}{x}) d x

\int 1 - \frac{1}{x} d x

\int \frac{x - 2}{x ( x ^{2} - x - 6 )} d x

\int (\frac{x}{x + 2} + x) d x

What is the integral of ((e^x)/(e^x+5)) ?
The integral of ((e^x)/(e^x+5)) is ln|e^x+5|+C