What is the integral of (5^{ln(x^3)})/(4x) ?

The integral of (5^{ln(x^3)})/(4x) is (125^{ln(x)})/(12ln(5))+C

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integral of (5^{ln(x^3)})/(4x)

\int \frac{5 ^{l n (x^{3})}}{4 x} d x

\frac{12 5 ^{l n (x)}}{12 ln ( 5 )} + C

Solution steps

\int \frac{5 ^{l n (x^{3})}}{4 x} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{5 ^{l n (x^{3})}}{x} d x

Simplify

\frac{5 ^{l n (x^{3})}}{x} : \frac{12 5 ^{l n (x)}}{x}

= \frac{1}{4} \cdot \int \frac{12 5 ^{l n (x)}}{x} d x

Apply u-substitution

: \int 12 5^{u} d u

= \frac{1}{4} \cdot \int 12 5^{u} d u

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

= \frac{1}{4} \cdot \frac{12 5 ^{u}}{ln ( 125 )}

Substitute back

u = ln (x)

= \frac{1}{4} \cdot \frac{12 5 ^{l n (x)}}{ln ( 125 )}

Simplify

\frac{1}{4} \cdot \frac{12 5 ^{l n (x)}}{ln ( 125 )} : \frac{12 5 ^{l n (x)}}{12 ln ( 5 )}

= \frac{12 5 ^{l n (x)}}{12 ln ( 5 )}

Add a constant to the solution

= \frac{12 5 ^{l n (x)}}{12 ln ( 5 )} + C

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What is the integral of (5^{ln(x^3)})/(4x) ?
The integral of (5^{ln(x^3)})/(4x) is (125^{ln(x)})/(12ln(5))+C