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

The integral of (3-x)/((3x+1)(x-1)) is -5/6 ln|3x+1|+1/2 ln|x-1|+C

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

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

- \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{3 - x}{( 3 x + 1 ) ( x - 1 )} : - \frac{5}{2 ( 3 x + 1 )} + \frac{1}{2 ( x - 1 )}

= \int - \frac{5}{2 ( 3 x + 1 )} + \frac{1}{2 ( 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{5}{2 ( 3 x + 1 )} d x + \int \frac{1}{2 ( x - 1 )} d x

\int \frac{5}{2 ( 3 x + 1 )} d x = \frac{5}{6} ln ∣ 3 x + 1 ∣

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

= - \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣

Add a constant to the solution

= - \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + C

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What is the integral of (3-x)/((3x+1)(x-1)) ?
The integral of (3-x)/((3x+1)(x-1)) is -5/6 ln|3x+1|+1/2 ln|x-1|+C