Question

How do you integrate `(5x)/(2x^2+11x+12)` using partial fractions?

Answer 1

`int((5x)/(2x^2 + 11x + 12))dx = 4ln|x + 4| - 3/2ln|2x + 3| + C`

Explanation:

We start by factoring the denominator.

`2x^2 + 11x + 12 = 2x^2 + 8x + 3x + 12 = 2x(x + 4) + 3(x +4) = (2x + 3)(x + 4)`.

The partial fraction decomposition will therefore be of the form:

`A/(2x + 3) + B/(x + 4) = (5x)/((2x + 3)(x + 4)`

`A(x + 4) + B(2x + 3) = 5x`

`Ax + 4A + 2Bx+ 3B = 5x`

`(A + 2B)x + (4A + 3B) = 5x`

We now write a systems of equations.

`{(A + 2B= 5), (4A + 3B = 0):}`

Solve:

`A = 5 - 2B`

`4(5 - 2B) + 3B = 0`

`20 - 8B + 3B = 0`

`-5B = -20`

`B = 4`

`A + 2(4) = 5`

`A = -3`

Hence, the partial fraction decomposition is `4/(x + 4) - 3/(2x + 3)`. The integral becomes

`int(4/(x + 4) - 3/(2x + 3))dx`

We know that `int(1/u)du = ln|u| + C`. Therefore:

`int(4/(x + 4) - 3/(2x + 3))dx = 4ln|x + 4| - 3/2ln|2x + 3| + C`

Hopefully this helps!