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

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

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integral of (2^{sqrt(x)})/(sqrt(x))

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) - l n ( 2 )}{l n ( 2 )}}} d u

= \int \frac{u}{ln ( 2 ) \cdot 2 ^{\frac{l n ( u ) - l n ( 2 )}{l n ( 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{u}{2 ^{\frac{l n ( u ) - l n ( 2 )}{l n ( 2 )}}} d u

Apply Integration By Parts

: \frac{u ^{2}}{2 ^{\frac{l n ( u )}{l n ( 2 )}}} - \int - 1 d u

= \frac{1}{ln ( 2 )} (\frac{u ^{2}}{2 ^{\frac{l n ( u )}{l n ( 2 )}}} - \int - 1 d u)

\int - 1 d u = - u

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

Substitute back

u = 2^{x}

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

Simplify

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

= \frac{2 \cdot 2 ^{x}}{ln ( 2 )}

Add a constant to the solution

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

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