What is the integral of (e^{5x})/(2+e^{5x)} ?

The integral of (e^{5x})/(2+e^{5x)} is 1/5 ln|2+e^{5x}|+C

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integral of (e^{5x})/(2+e^{5x)}

\int \frac{e ^{5 x}}{2 + e ^{5 x}} d x

\frac{1}{5} ln 2 + e^{5 x} + C

Solution steps

\int \frac{e ^{5 x}}{2 + e ^{5 x}} d x

Apply u-substitution

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

= \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 = 2 + e^{5 x}

= \frac{1}{5} ln 2 + e^{5 x}

Add a constant to the solution

= \frac{1}{5} ln 2 + e^{5 x} + C

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What is the integral of (e^{5x})/(2+e^{5x)} ?
The integral of (e^{5x})/(2+e^{5x)} is 1/5 ln|2+e^{5x}|+C