What is the integral of (9x^3-9x+2)/(x^3-x^2) ?

The integral of (9x^3-9x+2)/(x^3-x^2) is 9x+7ln|x|+2/x+2ln|x-1|+C

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integral of (9x^3-9x+2)/(x^3-x^2)

\int \frac{9 x ^{3} - 9 x + 2}{x ^{3} - x ^{2}} d x

9 x + 7 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{9 x ^{3} - 9 x + 2}{x ^{3} - x ^{2}} d x

Take the partial fraction of

\frac{9 x ^{3} - 9 x + 2}{x ^{3} - x ^{2}} : 9 + \frac{7}{x} - \frac{2}{x ^{2}} + \frac{2}{x - 1}

= \int 9 + \frac{7}{x} - \frac{2}{x ^{2}} + \frac{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 9 d x + \int \frac{7}{x} d x - \int \frac{2}{x ^{2}} d x + \int \frac{2}{x - 1} d x

\int 9 d x = 9 x

\int \frac{7}{x} d x = 7 ln ∣ x ∣

\int \frac{2}{x ^{2}} d x = - \frac{2}{x}

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

= 9 x + 7 ln ∣ x ∣ - (- \frac{2}{x}) + 2 ln ∣ x - 1 ∣

Simplify

= 9 x + 7 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= 9 x + 7 ln ∣ x ∣ + \frac{2}{x} + 2 ln ∣ x - 1 ∣ + C

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What is the integral of (9x^3-9x+2)/(x^3-x^2) ?
The integral of (9x^3-9x+2)/(x^3-x^2) is 9x+7ln|x|+2/x+2ln|x-1|+C