What is the integral of (5e^x)/(1+3e^x) ?

The integral of (5e^x)/(1+3e^x) is 5/3 ln|1+3e^x|+C

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integral of (5e^x)/(1+3e^x)

\int \frac{5 e ^{x}}{1 + 3 e ^{x}} d x

\frac{5}{3} ln ∣ 1 + 3 e^{x} ∣ + C

Solution steps

\int \frac{5 e ^{x}}{1 + 3 e ^{x}} d x

Take the constant out:

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

= 5 \cdot \int \frac{e ^{x}}{1 + 3 e ^{x}} d x

Apply u-substitution

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

= 5 \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

= 5 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 1 + 3 e^{x}

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

Simplify

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

= \frac{5}{3} ln ∣ 1 + 3 e^{x} ∣

Add a constant to the solution

= \frac{5}{3} ln ∣ 1 + 3 e^{x} ∣ + C

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What is the integral of (5e^x)/(1+3e^x) ?
The integral of (5e^x)/(1+3e^x) is 5/3 ln|1+3e^x|+C