Question

How do you integrate `(x^3+1)/((x^2-1)^2(x+1))` using partial fractions?

Answer 1

Rewrite the denominator as `(x + 1)³(x - 1)²` and then proceed with the normal expansion process

Explanation:

`(x³ + 1)/((x + 1)³(x - 1)²)` =
`A/((x + 1)) + B/((x + 1)²) + C/((x + 1)³) + D/(x - 1) + E/((x - 1)²`

Multiply both sides by the denominator:

`x³ + 1 = A(x + 1)²(x - 1)² + B(x + 1)(x - 1)² + C(x - 1)² + D(x + 1)³(x - 1) + E(x + 1)³`

Let x = -1:

`-1³ + 1 = C(-1 - 1)²`
`0 = C(-2)²`
`C = 0`

Let x = 1:

`1³ + 1 = E(1 + 1)³`
`2 = E(2)³`
`E = 1/4`

If you continue by creating 3 equations to solve for the remaining 3 unknow values, you will find that:

`A = 0, B = 3/4 and D = 0`

This gives you two integrals:

`3/4int 1/((x + 1)²)dx + 1/4int 1/((x - 1)²)dx`

Both integrals are trivial and found in lists of integrals.