What is the integral of 1/6 ln(1/3 x) ?

The integral of 1/6 ln(1/3 x) is 1/6 (xln(x/3)-x)+C

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integral of 1/6 ln(1/3 x)

\int \frac{1}{6} ln (\frac{1}{3} x) d x

\frac{1}{6} (x ln (\frac{x}{3}) - x) + C

Solution steps

\int \frac{1}{6} ln (\frac{1}{3} x) d x

Take the constant out:

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

= \frac{1}{6} \cdot \int ln (\frac{1}{3} x) d x

Simplify

\frac{1}{3} x : \frac{x}{3}

= \frac{1}{6} \cdot \int ln (\frac{x}{3}) d x

Apply Integration By Parts

: x ln (\frac{x}{3}) - \int 1 d x

= \frac{1}{6} (x ln (\frac{x}{3}) - \int 1 d x)

\int 1 d x = x

= \frac{1}{6} (x ln (\frac{x}{3}) - x)

Add a constant to the solution

= \frac{1}{6} (x ln (\frac{x}{3}) - x) + C

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What is the integral of 1/6 ln(1/3 x) ?
The integral of 1/6 ln(1/3 x) is 1/6 (xln(x/3)-x)+C