What is the integral of 5^{sqrt(x)} 1/(sqrt(x)) ?

The integral of 5^{sqrt(x)} 1/(sqrt(x)) is (2*5^{sqrt(x)})/(ln(5))+C

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integral of 5^{sqrt(x)} 1/(sqrt(x))

\int 5^{x} \frac{1}{x} d x

\frac{2 \cdot 5 ^{x}}{ln ( 5 )} + C

Solution steps

\int 5^{x} \cdot \frac{1}{x} d x

Simplify

5^{x} \cdot \frac{1}{x} : \frac{5 ^{x}}{x}

= \int \frac{5 ^{x}}{x} d x

Apply u-substitution

: \int 2 \cdot 5^{u} d u

= \int 2 \cdot 5^{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int 5^{u} d u

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

= 2 \cdot \frac{5 ^{u}}{ln ( 5 )}

Substitute back

u = x

= 2 \cdot \frac{5 ^{x}}{ln ( 5 )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 5 ^{x}}{ln ( 5 )}

Add a constant to the solution

= \frac{2 \cdot 5 ^{x}}{ln ( 5 )} + C

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What is the integral of 5^{sqrt(x)} 1/(sqrt(x)) ?
The integral of 5^{sqrt(x)} 1/(sqrt(x)) is (2*5^{sqrt(x)})/(ln(5))+C