What is the integral of (e^{5x})/(e^{5x)+9} ?

The integral of (e^{5x})/(e^{5x)+9} is 1/5 ln|e^{5x}+9|+C

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integral of (e^{5x})/(e^{5x)+9}

\int \frac{e ^{5 x}}{e ^{5 x} + 9} d x

\frac{1}{5} ln e^{5 x} + 9 + C

Solution steps

\int \frac{e ^{5 x}}{e ^{5 x} + 9} d x

Apply u-substitution

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = e^{5 x} + 9

= \frac{1}{5} ln e^{5 x} + 9

Add a constant to the solution

= \frac{1}{5} ln e^{5 x} + 9 + C

What is the integral of (e^{5x})/(e^{5x)+9} ?
The integral of (e^{5x})/(e^{5x)+9} is 1/5 ln|e^{5x}+9|+C