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

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

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

\int \frac{1}{x ( 1 + 3 x )} d x

ln ∣ x ∣ - ln ∣ 3 x + 1 ∣ + C

Solution steps

\int \frac{1}{x ( 1 + 3 x )} d x

Take the partial fraction of

\frac{1}{x ( 1 + 3 x )} : \frac{1}{x} - \frac{3}{3 x + 1}

= \int \frac{1}{x} - \frac{3}{3 x + 1} 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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{3}{3 x + 1} d x = ln ∣ 3 x + 1 ∣

= ln ∣ x ∣ - ln ∣ 3 x + 1 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ 3 x + 1 ∣ + C

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