What is the integral of (e^θ)/(4+3e^θ) ?

The integral of (e^θ)/(4+3e^θ) is 1/3 ln|4+3e^θ|+C

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integral of (e^θ)/(4+3e^θ)

\int \frac{e ^{θ}}{4 + 3 e ^{θ}} d θ

\frac{1}{3} ln 4 + 3 e^{θ} + C

Solution steps

\int \frac{e ^{θ}}{4 + 3 e ^{θ}} d θ

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 4 + 3 e^{θ}

= \frac{1}{3} ln 4 + 3 e^{θ}

Add a constant to the solution

= \frac{1}{3} ln 4 + 3 e^{θ} + C

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What is the integral of (e^θ)/(4+3e^θ) ?
The integral of (e^θ)/(4+3e^θ) is 1/3 ln|4+3e^θ|+C