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

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

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

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

\frac{1}{ln ( 2 )} ln ∣ 2^{x} + 3 ∣ + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{ln ( 2 ) u} d u

Take the constant out:

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

= \frac{1}{ln ( 2 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{ln ( 2 )} ln ∣ u ∣

Substitute back

u = 2^{x} + 3

= \frac{1}{ln ( 2 )} ln ∣ 2^{x} + 3 ∣

Add a constant to the solution

= \frac{1}{ln ( 2 )} ln ∣ 2^{x} + 3 ∣ + C

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