Question

How do you use partial fractions to find the integral `int 3/(x^2+x-2) dx`?

Answer 1

The integral equals `ln|x - 1| - ln|x + 2| + C`

Explanation:

Note that `x^2 + x - 2` can be factored as `(x + 2)(x - 1)`.

Thus , our partial fraction decomposition will be of the form

`A/(x + 2) + B/(x - 1) = 3/((x +2)(x - 1))`

`A(x - 1) + B(x + 2) = 3`

`Ax- A + Bx + 2B = 3`

`(A + B)x + (2B - A) = 3`

Write a system of equations and solve:

`{(A + B = 0), (2B - A = 3):}`

Solve either through substitution or elimination to get `A = -1` and `B = 1`.

Therefore, the integral becomes

`int1/(x- 1) - 1/(x + 2)dx`

This can be solved using `int1/xdx = ln|x| + C`.

`=ln|x - 1| - ln|x + 2| + C`

Hopefully this helps!