Calculus of a Single Variable, Chapter 5, 5.2, Section 5.2, Problem 19

In Substitution Rule, we follow int f(g(x))g'(x) dx = int f(u) du where we let u = g(x) .
Before we use this, we look for possible way to simplify the function using math operation or algebraic techniques.
For the problem: int (x^4+x-4)/(x^2+2) dx ,we expand first using long division.
(x^4+x-4)/(x^2+2) = x^2-2+x/(x^2+2)
Applying int (f(x) +- g(x))dx = int f(x) dx +- intg(x)dx :

We get int x^2 dx - int 2 dx + int x/(x^2+2) dx.
int x^2 dx = x^3/3
int 2 dx =2x
int x/(x^2+2) dx = 1/2 ln|x^2+2|
We use u-substitution on int x/(x^2+2) dx by letting u = x^2 +2
then du = 2x *dx rearrange into x* dx= (du)/2
Substituting u =x^2+2 and x * dx = (du)/2, the integral becomes:
int x/(x^2+2) dx = int 1/u *(du)/2
= 1/2 int (du)/u
= 1/2 ln|u|
Substitute u=x^2+2 then int 1/2 ln|u| = 1/2ln |x^2+2|
int x^2 dx - int 2 dx + int x/(x^2+2)dx = x^3/3 - 2x+1/2ln|x^2+2| +C