What is the integral of (x^2+11x-30)/(x^3-5x^2+6x) ?

The integral of (x^2+11x-30)/(x^3-5x^2+6x) is -5ln|x|+2ln|x-2|+4ln|x-3|+C

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integral of (x^2+11x-30)/(x^3-5x^2+6x)

\int \frac{x ^{2} + 11 x - 30}{x ^{3} - 5 x ^{2} + 6 x} d x

- 5 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣ + 4 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{2} + 11 x - 30}{x ^{3} - 5 x ^{2} + 6 x} d x

Take the partial fraction of

\frac{x ^{2} + 11 x - 30}{x ^{3} - 5 x ^{2} + 6 x} : - \frac{5}{x} + \frac{2}{x - 2} + \frac{4}{x - 3}

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

\int \frac{4}{x - 3} d x = 4 ln ∣ x - 3 ∣

= - 5 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣ + 4 ln ∣ x - 3 ∣

Add a constant to the solution

= - 5 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣ + 4 ln ∣ x - 3 ∣ + C

What is the integral of (x^2+11x-30)/(x^3-5x^2+6x) ?
The integral of (x^2+11x-30)/(x^3-5x^2+6x) is -5ln|x|+2ln|x-2|+4ln|x-3|+C