What is the integral of (2log_{4}(x))/x ?

The integral of (2log_{4}(x))/x is 2(log_{4}(x)ln|x|-(ln|x|^2)/(4ln(2)))+C

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integral of (2log_{4}(x))/x

\int \frac{2 lo g _{4} ( x )}{x} d x

2 (lo g_{4} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{4 ln ( 2 )}) + C

Solution steps

\int \frac{2 lo g _{4} ( x )}{x} d x

Take the constant out:

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

= 2 \cdot \int \frac{lo g _{4} ( x )}{x} d x

Apply Integration By Parts

: lo g_{4} (x) ln ∣ x ∣ - \int \frac{ln ∣ x ∣}{2 ln ( 2 ) x} d x

= 2 (lo g_{4} (x) ln ∣ x ∣ - \int \frac{ln ∣ x ∣}{2 ln ( 2 ) x} d x)

\int \frac{ln ∣ x ∣}{2 ln ( 2 ) x} d x = \frac{ln ∣ x ∣ ^{2}}{4 ln ( 2 )}

= 2 (lo g_{4} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{4 ln ( 2 )})

Add a constant to the solution

= 2 (lo g_{4} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{4 ln ( 2 )}) + C

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What is the integral of (2log_{4}(x))/x ?
The integral of (2log_{4}(x))/x is 2(log_{4}(x)ln|x|-(ln|x|^2)/(4ln(2)))+C