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

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

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integral of (3x-1)/(3x)

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

x - \frac{1}{3} ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 3 d x = 3 x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{1}{3} (3 x - ln ∣ x ∣)

Simplify

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

= x - \frac{1}{3} ln ∣ x ∣

Add a constant to the solution

= x - \frac{1}{3} ln ∣ x ∣ + C

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