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

The integral of (log_{3}(x))/(3x) is 1/3 (log_{3}(x)ln|x|-(ln|x|^2)/(2ln(3)))+C

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

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

\frac{1}{3} (lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )}) + C

Solution steps

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

Take the constant out:

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

= \frac{1}{3} \cdot \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

= \frac{1}{3} (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 )}

= \frac{1}{3} (lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )})

Add a constant to the solution

= \frac{1}{3} (lo g_{3} (x) ln ∣ x ∣ - \frac{ln ∣ x ∣ ^{2}}{2 ln ( 3 )}) + C

\int sin (4 x) cos^{2} (2 x) sin (x) d x

tan (3) x^{2}

\frac{\partial}{\partial x} (y^{x^{2} - y})

3 t - 4 t^{2}

\frac{d y}{d x} y = cos^{- 1} (\frac{1}{x})

What is the integral of (log_{3}(x))/(3x) ?
The integral of (log_{3}(x))/(3x) is 1/3 (log_{3}(x)ln|x|-(ln|x|^2)/(2ln(3)))+C