What is the integral of x/((3x-2)(2x+1)) ?

The integral of x/((3x-2)(2x+1)) is 2/21 ln|3x-2|+1/14 ln|2x+1|+C

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integral of x/((3x-2)(2x+1))

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

\frac{2}{21} ln ∣ 3 x - 2 ∣ + \frac{1}{14} ln ∣ 2 x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x}{( 3 x - 2 ) ( 2 x + 1 )} : \frac{2}{7 ( 3 x - 2 )} + \frac{1}{7 ( 2 x + 1 )}

= \int \frac{2}{7 ( 3 x - 2 )} + \frac{1}{7 ( 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{2}{7 ( 3 x - 2 )} d x + \int \frac{1}{7 ( 2 x + 1 )} d x

\int \frac{2}{7 ( 3 x - 2 )} d x = \frac{2}{21} ln ∣ 3 x - 2 ∣

\int \frac{1}{7 ( 2 x + 1 )} d x = \frac{1}{14} ln ∣ 2 x + 1 ∣

= \frac{2}{21} ln ∣ 3 x - 2 ∣ + \frac{1}{14} ln ∣ 2 x + 1 ∣

Add a constant to the solution

= \frac{2}{21} ln ∣ 3 x - 2 ∣ + \frac{1}{14} ln ∣ 2 x + 1 ∣ + C

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What is the integral of x/((3x-2)(2x+1)) ?
The integral of x/((3x-2)(2x+1)) is 2/21 ln|3x-2|+1/14 ln|2x+1|+C