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

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

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

\int \frac{1}{2 ^{l n (x)}} d x

\frac{x}{( 1 - ln ( 2 ) ) \cdot 2 ^{l n (x)}} + C

Solution steps

\int \frac{1}{2 ^{l n (x)}} d x

Apply u-substitution

: \int (\frac{e}{2})^{u} d u

= \int (\frac{e}{2})^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{( \frac{e}{2} ) ^{u}}{ln ( \frac{e}{2} )}

Substitute back

u = ln (x)

= \frac{( \frac{e}{2} ) ^{l n (x)}}{ln ( \frac{e}{2} )}

Simplify

\frac{( \frac{e}{2} ) ^{l n (x)}}{ln ( \frac{e}{2} )} : \frac{x}{( 1 - ln ( 2 ) ) \cdot 2 ^{l n (x)}}

= \frac{x}{( 1 - ln ( 2 ) ) \cdot 2 ^{l n (x)}}

Add a constant to the solution

= \frac{x}{( 1 - ln ( 2 ) ) \cdot 2 ^{l n (x)}} + C

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