Separation of integral by approximation

Question

I'm working with the following integral $\displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}}$ and would like to split it in something like $$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x \sqrt{1-bx^2}}$$ or something like that (separate the integrals in one that only depends on $a$ and one that only depends on $b$, but I definitely want to keep the $a$ dependent one as I've written it!). Is that possible? I don't need anything exact, approximations work for me too!!!

Answer 1

The problem is that the indefinite integral $\int \dfrac{dx}{x\sqrt{1-ax-bx^2}}$ is $\ln(x)-\ln(2\sqrt{1-ax-bx^2}-ax+2)+C$.

With the bounds provided in the problem, the integral would not converge as $\lim_{x\rightarrow 0} \ln(x) =-\infty $

Answer 2

For $x$ close to $0$, we have $\sqrt{1-ax-bx^2}$ close to $1$ and therefore $<2$. Consequently for $y>0$ we have $$ \int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}} \ge \int_0^y \frac {dx}{x\cdot 2} = +\infty. $$

If you had $\displaystyle\int_1^y$ instead of $\displaystyle\int_0^y$ then it would be a harder problem: we'd have to find an antiderivative of the function begin integrated.