What is the integral of (2e^x)/(3e^x+1) ?

The integral of (2e^x)/(3e^x+1) is 2/3 ln|3e^x+1|+C

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integral of (2e^x)/(3e^x+1)

\int \frac{2 e ^{x}}{3 e ^{x} + 1} d x

\frac{2}{3} ln ∣ 3 e^{x} + 1 ∣ + C

Solution steps

\int \frac{2 e ^{x}}{3 e ^{x} + 1} d x

Take the constant out:

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

= 2 \cdot \int \frac{e ^{x}}{3 e ^{x} + 1} d x

Apply u-substitution

= 2 \cdot \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

= 2 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 3 e^{x} + 1

= 2 \cdot \frac{1}{3} ln ∣ 3 e^{x} + 1 ∣

Simplify

2 \cdot \frac{1}{3} ln ∣ 3 e^{x} + 1 ∣ : \frac{2}{3} ln ∣ 3 e^{x} + 1 ∣

= \frac{2}{3} ln ∣ 3 e^{x} + 1 ∣

Add a constant to the solution

= \frac{2}{3} ln ∣ 3 e^{x} + 1 ∣ + C

What is the integral of (2e^x)/(3e^x+1) ?
The integral of (2e^x)/(3e^x+1) is 2/3 ln|3e^x+1|+C