What is the integral of (2^x)/(2^x(1+4^x)) ?

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

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integral of (2^x)/(2^x(1+4^x))

integral

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

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

Solution steps

\int \frac{2 ^{x}}{2 ^{x} ( 1 + 4 ^{x} )} d x

Cancel the common factor:

2^{x}

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

\frac{1}{1 + 4 ^{x}} = - \frac{4 ^{x}}{1 + 4 ^{x}} + 1

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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