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

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Answer 1 · 2016-10-30T20:54:17

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

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

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

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

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