What is the integral of (8e^{8s})/(3+e^{8s)} ?

The integral of (8e^{8s})/(3+e^{8s)} is ln|3+e^{8s}|+C

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integral of (8e^{8s})/(3+e^{8s)}

\int \frac{8 e ^{8 s}}{3 + e ^{8 s}} d s

ln 3 + e^{8 s} + C

Solution steps

\int \frac{8 e ^{8 s}}{3 + e ^{8 s}} d s

Take the constant out:

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

= 8 \cdot \int \frac{e ^{8 s}}{3 + e ^{8 s}} d s

Apply u-substitution

: \int \frac{1}{8 u} d u

= 8 \cdot \int \frac{1}{8 u} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 3 + e^{8 s}

= 8 \cdot \frac{1}{8} ln 3 + e^{8 s}

Simplify

8 \cdot \frac{1}{8} ln 3 + e^{8 s} : ln 3 + e^{8 s}

= ln 3 + e^{8 s}

Add a constant to the solution

= ln 3 + e^{8 s} + C

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What is the integral of (8e^{8s})/(3+e^{8s)} ?
The integral of (8e^{8s})/(3+e^{8s)} is ln|3+e^{8s}|+C