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

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

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integral of (-1)/(3x)

\int \frac{- 1}{3 x} d x

- \frac{1}{3} ln ∣ x ∣ + C

Solution steps

\int \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}{3} \cdot \int \frac{1}{x} d x

Simplify

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

Use the common integral:

\int \frac{1}{x} d x = ln (∣ x ∣)

= (- \frac{1}{3}) ln ∣ x ∣

Simplify

= - \frac{1}{3} ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ x ∣ + C