Question

How do you use partial fractions to find the integral `int (x^2-1)/(x^3+x)dx`?

Answer 1

`ln|(x^2 + 1)/x|+ C`

Explanation:

Start by factoring the denominator.

`x^3 + x = x(x^2 + 1)`

Now write the partial fraction decomposition.

`(Ax + B)/(x^2 + 1) + C/x = (x^2 - 1)/(x(x^2 + 1))`

`(Ax + B)x + C(x^2 + 1) = x^2 - 1`

`Ax^2 + Bx + Cx^2 + C = x^2 - 1`

`(A + C)x^2 + Bx + C = x^2 - 1`

We now write a system of equations.

`{(A + C = 1), (B = 0), (C = -1):}`

Solving, we get `A = 2, B = 0, C = -1`.

`int(2x)/(x^2 + 1) - 1/xdx`

`int (2x)/(x^2 + 1)dx - int 1/xdx`

We now make the substitution `u = x^2 + 1`. Then `du = 2xdx` and `dx = (du)/(2x)`.

`int (2x)/u * (du)/(2x) - int 1/xdx`

`int 1/u du - int 1/x dx`

`ln|u| - ln|x| + C`

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

`ln|(x^2 + 1)/x|+ C`

Hopefully this helps!