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

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

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

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

2 ln ∣ 3 + e^{x} ∣ + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 3 + e^{x}

= 2 ln ∣ 3 + e^{x} ∣

Add a constant to the solution

= 2 ln ∣ 3 + e^{x} ∣ + C

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