What is the integral of (-5x^2+11x-10)/(x^3-x^2-6x) ?

The integral of (-5x^2+11x-10)/(x^3-x^2-6x) is 5/3 ln|x|-22/15 ln|x-3|-26/5 ln|x+2|+C

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integral of (-5x^2+11x-10)/(x^3-x^2-6x)

\int \frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} d x

\frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} d x

Expand

\frac{- 5 x ^{2} + 11 x - 10}{x ^{3} - x ^{2} - 6 x} : - \frac{5 x ^{2}}{x ^{3} - x ^{2} - 6 x} + \frac{11 x}{x ^{3} - x ^{2} - 6 x} - \frac{10}{x ^{3} - x ^{2} - 6 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 ^{2}}{x ^{3} - x ^{2} - 6 x} d x + \int \frac{11 x}{x ^{3} - x ^{2} - 6 x} d x - \int \frac{10}{x ^{3} - x ^{2} - 6 x} d x

\int \frac{5 x ^{2}}{x ^{3} - x ^{2} - 6 x} d x = 2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣

\int \frac{11 x}{x ^{3} - x ^{2} - 6 x} d x = 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣)

\int \frac{10}{x ^{3} - x ^{2} - 6 x} d x = - \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣

= - (2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣) + 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - (- \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣)

Simplify

- (2 ln ∣ x + 2 ∣ + 3 ln ∣ x - 3 ∣) + 11 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - (- \frac{5}{3} ln ∣ x ∣ + ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 3 ∣) : \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣

= \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{5}{3} ln ∣ x ∣ - \frac{22}{15} ln ∣ x - 3 ∣ - \frac{26}{5} ln ∣ x + 2 ∣ + C

\frac{40 x ^{5}}{e ^{x}}

\frac{d}{d x} (\frac{2}{x - 7})

(4 x y + 3 y^{2} - x) d x + (x^{2} + 2 x y) d y = 0

y = \frac{x ^{2}}{x + 4}, (3, \frac{9}{7})

\int_{- 7}^{7} (49 - x^{2})^{\frac{1}{2}} d x

What is the integral of (-5x^2+11x-10)/(x^3-x^2-6x) ?
The integral of (-5x^2+11x-10)/(x^3-x^2-6x) is 5/3 ln|x|-22/15 ln|x-3|-26/5 ln|x+2|+C