Question

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

Answer 1

By converting the fraction into a sum of independent factors

Explanation:

We know how to integrate functions of the form `1/x^n` or `1/(x-a)^n` so we want to transform the complicated fraction into a sum of simple fractions. We write
`I = A/(x-1) + B/(x-1)^2 + C/(x^2 + 1)`
The constant term in the numerator is `B + C-A` and is equal to `-1`
The `x^4` term is `B+C+A` and must be zero. This gives us
`-2A = -1` or `A=-1/2`
The `x^3` term is `-2C + A + B = 0`

Next `-2C + A +B +2C = 0 + 2(-A-B)` so
`B = -A = 1/2` and thus `C=0`

Thus `I = -(1/2)1/(x-1) + (1/2)1/(x-1)^2`

Integral `= -1/2 ln(x-1) -1/2(x-1)^2`