What is the integral of (5x^4+9)/(x^2-x-6) ?

The integral of (5x^4+9)/(x^2-x-6) is (5x^3)/3+(5x^2)/2+35x-89/5 ln|x+2|+414/5 ln|x-3|+C

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integral of (5x^4+9)/(x^2-x-6)

\int \frac{5 x ^{4} + 9}{x ^{2} - x - 6} d x

\frac{5 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 35 x - \frac{89}{5} ln ∣ x + 2 ∣ + \frac{414}{5} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{5 x ^{4} + 9}{x ^{2} - x - 6} d x

Long division

\frac{5 x ^{4} + 9}{x ^{2} - x - 6} : 5 x^{2} + 5 x + 35 + \frac{65 x + 219}{x ^{2} - x - 6}

= \int 5 x^{2} + 5 x + 35 + \frac{65 x + 219}{x ^{2} - x - 6} 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 5 x^{2} d x + \int 5 x d x + \int 35 d x + \int \frac{65 x + 219}{x ^{2} - x - 6} d x

\int 5 x^{2} d x = \frac{5 x ^{3}}{3}

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 35 d x = 35 x

\int \frac{65 x + 219}{x ^{2} - x - 6} d x = - \frac{89}{5} ln ∣ x + 2 ∣ + \frac{414}{5} ln ∣ x - 3 ∣

= \frac{5 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 35 x - \frac{89}{5} ln ∣ x + 2 ∣ + \frac{414}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{5 x ^{3}}{3} + \frac{5 x ^{2}}{2} + 35 x - \frac{89}{5} ln ∣ x + 2 ∣ + \frac{414}{5} ln ∣ x - 3 ∣ + C

What is the integral of (5x^4+9)/(x^2-x-6) ?
The integral of (5x^4+9)/(x^2-x-6) is (5x^3)/3+(5x^2)/2+35x-89/5 ln|x+2|+414/5 ln|x-3|+C