What is the integral of (15e^{5w})/(e^{5w)+2} ?

The integral of (15e^{5w})/(e^{5w)+2} is 3ln|e^{5w}+2|+C

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integral of (15e^{5w})/(e^{5w)+2}

\int \frac{15 e ^{5 w}}{e ^{5 w} + 2} d w

3 ln e^{5 w} + 2 + C

Solution steps

\int \frac{15 e ^{5 w}}{e ^{5 w} + 2} d w

Take the constant out:

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

= 15 \cdot \int \frac{e ^{5 w}}{e ^{5 w} + 2} d w

Apply u-substitution

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

= 15 \cdot \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

= 15 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 15 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = e^{5 w} + 2

= 15 \cdot \frac{1}{5} ln e^{5 w} + 2

Simplify

15 \cdot \frac{1}{5} ln e^{5 w} + 2 : 3 ln e^{5 w} + 2

= 3 ln e^{5 w} + 2

Add a constant to the solution

= 3 ln e^{5 w} + 2 + C

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What is the integral of (15e^{5w})/(e^{5w)+2} ?
The integral of (15e^{5w})/(e^{5w)+2} is 3ln|e^{5w}+2|+C