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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = ln u^{2} + 1 + u

= \frac{1}{ln ( 2 )} ln u^{2} + 1 + u

Substitute back

u = 2^{x}

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

(2^{x})^{2} = 4^{x}

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

Add a constant to the solution

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

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