Question on Integration solution I am looking through the solution of integrating: $$\int_{}^{} \frac{x^4+1}{x^3+1} dx$$
I understand it but for the following part:

My problem is limited to the first and penultimate column. I would very much appreciate it if someone could explain to me these, especially the penultimate one.
EDIT 0: Clarification
I have a problem with this line:

How do I get those thetas?
My main problem is with this part:

Why 1 = ..., -1 = ... and 0 = ...?

Solution of integral will be by partial fraction method
$$I=\int (x+\frac{1-x}{x^3+1})dx$$
$$I = \frac{x^2}{2}+\int (\frac{1-x}{(x+1)(x^2-x+1)})dx$$
Where you can solve for 
$$I_1=\int (\frac{1-x}{(x+1)(x^2-x+1)})dx$$
Using partial fraction like done here
https://www.mathsdiscussion.com/discussion-forum/topic/indefinite-integral/?part=1#postid-77
$$I=\frac{x^2}{2}-\frac{1}{3}ln(\frac{x^2-x+1}{(x+1)^2})+C$$
Problem what you say in your solution is when let
$$\frac{1-x}{(x+1)(x^2-x+1)}=\frac{a}{x+1}+\frac{bx+c}{x^2-x+1}$$
Multiplying both sides by $(x+1)(x^2-x+1)$ we ge
$$1-x=a(x^2-x+1)+(bx+c)(x+1)$$
Now comparing coefficients of $x^2$ , x and constant terms on both sides we get
$$  $$
Coefficient of$x^2$ on both sides 0=a+b
$$. $$
Coefficient of x on both sides -1=-a+b+c
$$. $$
Constant terms on both sides 1=a+c
$$ $$ This how we get those 3 equations
