integrating method (maybe PFD)

Question

I am trying to integrate:

dt = 1/(ax-bx^2) * dx

I am guessing I need to use Partial Fraction decomposition, can someone help show me how to begin this process?

Answer 1

To begin we have that:

$$dt= \frac{1}{ax-bx^2}dx$$

Now using the partial fraction decomp.

$$\frac{1}{ax-bx^2} = \frac{1}{x(a-bx)} = \frac{A}{x} + \frac{B}{a-bx}$$

Please note that $A$ doesn't equal $a$ and $B$ doesn't equal $b$.

Now from here we can multiply both sides by the LCM to get:

$$1 = (a-bx)A + Bx$$

Now if $x = 0$,

$$1 = aA + 0,\ A = \frac{1}{a}$$

If $x = \frac{a}{b}$ $$1 = 0*A+B*\frac{a}{b},\ B = \frac{b}{a}$$