The problem is to evaluate $\int \frac{5x^2+2x-5}{x^3-x}\,dx$.
This is the solution that I tried:
I really have no idea of this problem.
After check my solution, if there are any problem that I missed, or if approaching is wrong, please tell me how to approach this question.
$\endgroup$2 Answers
$\begingroup$A start: Use partial fractions. Find constants $A,B,C$ such that your function is equal to $\displaystyle \frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}$.
$\endgroup$ 1 $\begingroup$Hint:
$$\int\frac{5x^2+2x-5}{x^3-x}dx=\int\frac{5(x^2-1)+2x}{x(x^2-1)}dx=\int\frac{5(x^2-1)}{x(x^2-1)}dx+\int\frac{2}{x^2-1}dx=$$
$$=\int\frac{5}{x}dx+\int\frac{2}{x^2-1}dx= 5\int\frac{1}{x}dx+2\int\frac{1}{x^2-1}dx$$
The first is easy now, for the second you have to manipulate a little:
$$\int\frac{1}{x^2-1}dx=\int\frac{1}{(x+1)(x-1)}dx=\int(\frac{A}{x+1}+\frac{B}{x-1})dx$$
and find A and B.
$\endgroup$ 1
