Question

How do you integrate `(x^3-4x-10)/(x^2-x-6)` using partial fractions?

Answer 1

`ln|x-3| + 2ln|x+2| + c`

Explanation:

The first step is to factor the denominator of the function

`rArr (x^3-4x-10)/(x^2-x-6) = (x^3-4x-10)/((x-3)(x+2))`

since the factors are linear then the coefficients of the partial fractions will be constants , say A and B. Writing the function in terms of it's partial fractions.

`(x^3-4x-10)/((x-3)(x+2)) = A/(x-3) + B/(x+2)`

multiplying through by (x-3)(x+2)

`x^3-4x-10 = A(x+2) + B(x-3) ...................... (1)`

We now have to find the values of A and B .Note that if x = -2 the term with A will be zero and if x = 3 the term with B will be zero. We can make use of this fact in finding A and B.

let x = -2 in (1) :- 10 = -5B `rArrcolor(blue)" B = 2"`

let x = 3 in (1) : 5 = 5A `rArr color(blue)" A = 1 "`

`rArr (x^3-4x-10)/((x-3)(x+2)) = 1/(x-3) + 2/(x+2)`

Integral becomes : `int(dx)/(x-3) + int(2dx)/(x+2)`

`= ln|x-3| + 2ln|x+2| + c`