Question

Find the partial fractions decomposition and an antiderivative.

Answer 1

can be re-written as: \[\frac{ -3x-1 }{x(x^{2} - 1) }\] = \[\frac{ A }{ x } + \frac{ Bx + C }{ (x^{2} - 1) }\]

Answer 2

Why Bx + C you ask?
First of all, accept it, it's a rule when it's a second order polynomials.

But mechanically, notice that the numerator you need to match doesn't have a second order term (x^2). Therefore, when we'd cross multiply, the only way for the Ax^2 to dissapear would be to have a Bx^2 with B conveniently equal to A.

TL;DR. it's a rule, accept it :)

Answer 3

with B = -A of course since we're adding

Answer 4

ok

Answer 5

so what do i do next?

Answer 6

@experimentX

Answer 7

\[ \frac{ -3x-1 }{x(x^{2} - 1) } = \frac{-3x-1}{x(x+1)(x-1)} = \frac{A}{x} + \frac{B}{x-1}+\frac{C}{x+1}\]

Answer 8

take LCM and cross multiply.

Answer 9

umm how?

Answer 10

There is nice video lecture on Khan Academy
http://www.youtube.com/watch?v=S-XKGBesRzk

Answer 11

ok thx