What is the integral of (17x-3)/(3x^2+x-2) ?

The integral of (17x-3)/(3x^2+x-2) is 5/3 ln|3x-2|+4ln|x+1|+C

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integral of (17x-3)/(3x^2+x-2)

\int \frac{17 x - 3}{3 x ^{2} + x - 2} d x

\frac{5}{3} ln ∣ 3 x - 2 ∣ + 4 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{17 x - 3}{3 x ^{2} + x - 2} d x

Take the partial fraction of

\frac{17 x - 3}{3 x ^{2} + x - 2} : \frac{5}{3 x - 2} + \frac{4}{x + 1}

= \int \frac{5}{3 x - 2} + \frac{4}{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}{3 x - 2} d x + \int \frac{4}{x + 1} d x

\int \frac{5}{3 x - 2} d x = \frac{5}{3} ln ∣ 3 x - 2 ∣

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= \frac{5}{3} ln ∣ 3 x - 2 ∣ + 4 ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{5}{3} ln ∣ 3 x - 2 ∣ + 4 ln ∣ x + 1 ∣ + C

What is the integral of (17x-3)/(3x^2+x-2) ?
The integral of (17x-3)/(3x^2+x-2) is 5/3 ln|3x-2|+4ln|x+1|+C