Math Help - partial fractions

1. partial fractions

$\int\frac{x^2-x+1}{x^3+x}dx$

$\frac{x^2-x+1}{x(x^2+1)} = \frac{A}{x}+\frac{Bx+C}{x^2+1}$

$x^2-x+1 = A(x^2+1)+(Bx+C)X$

i plugged in $x=0$ and got $A=1$
need help finding $B$ and $C$

2. No, not quite. You got mixed up with $(x^{2}+1)^2$

$\frac{x^{2}-x+1}{x(x^{2}+1)}$

$\frac{A}{x}+\frac{B}{x^{2}+1}=x^{2}-x+1$

Now continue.

3. If you prefer, partial fractions can be avoid


\begin{aligned}
\int\frac{x^2-x+1}{x^3+x}~dx&=\int\frac{(x^2+1)-x}{x^3+x}~dx\\
&=\int\frac1{x}~dx-\int\frac1{1+x^2}~dx\\
&=\ln|x|-\arctan{x}+k,\color{blue}~k\in\mathbb{R}
\end{aligned}

I think partial fractions technique shoul be used if the problem really needs it.

4. Yes, I agree with Krizalid. Like the 'Hospital' rule. If I can, I prefer to evaluate limits without it first.