Need help with integral manipulation

Question

I have the following integral, I need to solve $$\int \frac{f'(t)}{af(t) - bf(t)^2}dt$$ where $a$ is a constant and $b \in (0,1)$. I used substitution. Let $u = f(t)$ and let $du = f'(t) dt$. I then get that $$\int \frac{du}{u(a - bu)}$$ I know the integral for $\int \frac{du}{u}$ but I dont know how to solve this one. Would it be correct if I said that $$\int \frac{du}{u(a - bu)} = \ln (u(a - bu)) + C $$

Answer 1

hint

To integrate a rational fraction of the form $$\frac{P(u)}{Q(u)}$$ with $ deg P<deg Q$ , we use partial fractions decomposition.

So

$$\frac{1}{u(a-bu)}=\frac Au + \frac{B}{a-bu}$$

with $$A=\frac 1a\text{ and } B=\frac ba$$

You should find that $$\int \frac{du}{u(a-bu)}=\frac 1a\ln(\Bigl|\frac{u}{a-bu}\Bigr|) +C$$

Answer 2

No, it's not correct. You have to proceed to a decomposition into partial fractions first: $$\frac 1{a(a-bu)}=\frac Au+ \frac B{a-bu}\qquad(A,B\in\mathbf R).$$