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

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

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

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

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

Solution steps

\int \frac{2 ^{x}}{1 + 2 ^{x}} 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 = 1 + 2^{x}

= \frac{1}{ln ( 2 )} ln ∣ 1 + 2^{x} ∣

Add a constant to the solution

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

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