What is the integral of (log_{3}(x))/x ?

The integral of (log_{3}(x))/x is log_{3}(x)ln|x|-(ln|x|^2)/(2ln(3))+C

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integral of (log_{3}(x))/x

\int \frac{lo g _{3} ( x )}{x} d x

lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )} + C

Solution steps

\int \frac{lo g _{3} ( x )}{x} d x

Apply Integration By Parts

: lo g_{3} (x) ln ∣ x ∣ - \int \frac{ln ∣ x ∣}{ln ( 3 ) x} d x

= lo g_{3} (x) ln ∣ x ∣ - \int \frac{ln ∣ x ∣}{ln ( 3 ) x} d x

\int \frac{ln ∣ x ∣}{ln ( 3 ) x} d x = \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )}

= lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )}

Add a constant to the solution

= lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )} + C

y^{^{'}} + (\frac{1 + x}{x}) y = 0

y = ln (x^{2}), (5, ln (25))

f (x) = 3 x^{8}

x \to 0 lim (x^{s i n (2 x)})

What is the integral of (log_{3}(x))/x ?
The integral of (log_{3}(x))/x is log_{3}(x)ln|x|-(ln|x|^2)/(2ln(3))+C