Question

How do you integrate `(x^3+x^2+2x+1)/((x^2+1)(x^2+2)) dx`?

Answer 1

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

Explanation:

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

There terms may be integrated by substitution.

For the first, use `u = x^2+1` to get `1/2ln (x^2+1)`.

For the second use `x=sqrt2u` to get `1/sqrt2 tan^-1(x/sqrt2)`

We can write the integral as

`1/2ln(x^2+1)+sqrt2/2tan^-1(x/sqrt2)`