What is the integral of (2e^x)/(5e^x+3) ?

The integral of (2e^x)/(5e^x+3) is 2/5 ln|5e^x+3|+C

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integral of (2e^x)/(5e^x+3)

\int \frac{2 e ^{x}}{5 e ^{x} + 3} d x

\frac{2}{5} ln ∣ 5 e^{x} + 3 ∣ + C

Solution steps

\int \frac{2 e ^{x}}{5 e ^{x} + 3} 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}}{5 e ^{x} + 3} d x

Apply u-substitution

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

= 2 \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

= 2 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 5 e^{x} + 3

= 2 \cdot \frac{1}{5} ln ∣ 5 e^{x} + 3 ∣

Simplify

2 \cdot \frac{1}{5} ln ∣ 5 e^{x} + 3 ∣ : \frac{2}{5} ln ∣ 5 e^{x} + 3 ∣

= \frac{2}{5} ln ∣ 5 e^{x} + 3 ∣

Add a constant to the solution

= \frac{2}{5} ln ∣ 5 e^{x} + 3 ∣ + C

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What is the integral of (2e^x)/(5e^x+3) ?
The integral of (2e^x)/(5e^x+3) is 2/5 ln|5e^x+3|+C