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

The integral of (2^{2x})/(1+2^{2x)} is 1/(2ln(2))ln|1+4^x|+C

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

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

\frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣ + C

Solution steps

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

Simplify

\frac{2 ^{2 x}}{1 + 2 ^{2 x}} : \frac{4 ^{x}}{1 + 4 ^{x}}

= \int \frac{4 ^{x}}{1 + 4 ^{x}} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2 ln ( 2 )} ln ∣ u ∣

Substitute back

u = 1 + 4^{x}

= \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣

Add a constant to the solution

= \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣ + C

t \frac{d y}{d t} + 4 y = t^{2}

x y^{2} y^{^{'}} = y^{3} - 5 x^{3}, y (1) = 2

f (x) = - x^{2} + 4

sin (x) + \frac{4}{5} cot (x)

f (x) = 3 x^{2} - 4 x + 1, at x = 2

What is the integral of (2^{2x})/(1+2^{2x)} ?
The integral of (2^{2x})/(1+2^{2x)} is 1/(2ln(2))ln|1+4^x|+C