<s>[INST] Why can we use the following integration method? In order to integrate the following function, our professor taught us the following method:
$\dfrac{x^4+1}{x^3+x^2} = \dfrac{x^4+1}{x^2(x+1)}=\dfrac{A}{x^2}+\dfrac{B}{x}+\dfrac{C}{x+1}$
I don't understand why we can do that, in reality, $\dfrac{x^4+1}{x^3+x^2} = \dfrac{x^4+1}{xx(x+1)}=\dfrac{A}{x}+\dfrac{B}{x}+\dfrac{C}{x+1}$ 
Although I understand why we cannot find $A, B$and$C$ that way (Indeed to find A we evaluate $\dfrac{x^4+1}{x^2(x+1)}$ at x=0)
 [/INST] You can see that your partial fraction decomposition cannot work simply noting that the numerator of the fraction: $A(x+1)+Bx(x+1)+Cx^2$ cannot be a polynomial of degree $4$. In other words, if the denominators are the factors of a polynomial of degree $n$, and the numerators of the partial fractions are numbers, that the numerator of their sum cannot be of degree $\ge n$.
As noted in the comments, the first step is to use long division to find:
$$
\frac{x^4+1}{x^3+x^2}=x-1+\frac{x^2+1}{x^3+x^2}
$$
than we can use partial fractions for:
$$
\frac{x^2+1}{x^2(x+1)}=\frac{A}{x^2}+\frac{B}{x}+\frac{C}{x+1}
$$
and find $A=1,B=-1,C=2$
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