What is the integral of ((5x-5))/(3x^2-8x-3) ?

The integral of ((5x-5))/(3x^2-8x-3) is 2/3 ln|3x+1|+ln|x-3|+C

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integral of ((5x-5))/(3x^2-8x-3)

\int \frac{( 5 x - 5 )}{3 x ^{2} - 8 x - 3} d x

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

Solution steps

\int \frac{5 x - 5}{3 x ^{2} - 8 x - 3} d x

Take the partial fraction of

\frac{5 x - 5}{3 x ^{2} - 8 x - 3} : \frac{2}{3 x + 1} + \frac{1}{x - 3}

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

\int \frac{2}{3 x + 1} d x = \frac{2}{3} ln ∣ 3 x + 1 ∣

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

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

Add a constant to the solution

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

What is the integral of ((5x-5))/(3x^2-8x-3) ?
The integral of ((5x-5))/(3x^2-8x-3) is 2/3 ln|3x+1|+ln|x-3|+C