What is the integral of (x+3)/(6x^3-7x^2-3x) ?

The integral of (x+3)/(6x^3-7x^2-3x) is -ln|x|+8/11 ln|3x+1|+3/11 ln|2x-3|+C

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integral of (x+3)/(6x^3-7x^2-3x)

\int \frac{x + 3}{6 x ^{3} - 7 x ^{2} - 3 x} d x

- ln ∣ x ∣ + \frac{8}{11} ln ∣ 3 x + 1 ∣ + \frac{3}{11} ln ∣ 2 x - 3 ∣ + C

Solution steps

\int \frac{x + 3}{6 x ^{3} - 7 x ^{2} - 3 x} d x

Take the partial fraction of

\frac{x + 3}{6 x ^{3} - 7 x ^{2} - 3 x} : - \frac{1}{x} + \frac{24}{11 ( 3 x + 1 )} + \frac{6}{11 ( 2 x - 3 )}

= \int - \frac{1}{x} + \frac{24}{11 ( 3 x + 1 )} + \frac{6}{11 ( 2 x - 3 )} 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{24}{11 ( 3 x + 1 )} d x + \int \frac{6}{11 ( 2 x - 3 )} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{24}{11 ( 3 x + 1 )} d x = \frac{8}{11} ln ∣ 3 x + 1 ∣

\int \frac{6}{11 ( 2 x - 3 )} d x = \frac{3}{11} ln ∣ 2 x - 3 ∣

= - ln ∣ x ∣ + \frac{8}{11} ln ∣ 3 x + 1 ∣ + \frac{3}{11} ln ∣ 2 x - 3 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{8}{11} ln ∣ 3 x + 1 ∣ + \frac{3}{11} ln ∣ 2 x - 3 ∣ + C

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What is the integral of (x+3)/(6x^3-7x^2-3x) ?
The integral of (x+3)/(6x^3-7x^2-3x) is -ln|x|+8/11 ln|3x+1|+3/11 ln|2x-3|+C