# Asymptotic behaviour of integral. How should I proceed?

Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $$b, \sigma: (l, r)\to\mathbb{R}$$, $$−\infty \leq l < r \leq \infty$$ bounded functions on compact intervals of $$(l, r)$$.
In particular, $$b(Y_t)=(u-(u+i)Y_t)$$ $$\sigma(Y_t)=o\sqrt{(Y_t)(1-Y_t)}$$ with $$u$$, $$i$$ and $$o$$ arbitrary parameters.
Hence, focus will be on the following SDE: $$dY_t=(u-(u+i)Y_t)dt+o\sqrt{(Y_t)(1-Y_t)}dW_t\tag{2}$$ I must check whether the process $$\{X_t\}$$ remains within the interval $$(l,r)$$ or not for each $$0\leq t\leq T$$.

To this, I use the Feller test for explosions. Such a test requires that the following two integrals must be defined and computed: $$p(x)=\int_c^x \exp\bigg\{-2\int_c^{\xi}\frac{b(\zeta)}{\sigma^2(\zeta)}d\zeta\bigg\}d\xi\tag{3}$$ $$v(x)=\int_c^x\frac{2(p(x)-p(y))}{p\hspace{0.1cm}'(y)\sigma^2(y)}dy\tag{4}$$ with $$c\in(l,r)$$.
According to Feller test, probability that the process at least touches the bounds of interval $$I$$ equals $$1$$ or is less than $$1$$ according to whether $$v(l+)=v(r-)=\infty$$ or not. Let us fix $$(l,r)=(0,1)$$ and $$c=\frac{1}{2}$$.

I would like to study the asymptotic behaviour of the integral $$(4)$$ with $$c=\frac{1}{2}$$ at bounds $$l=0$$ and $$r=1$$, but I have not any experience with analyses like that. Is there a good standard method or is it just a matter of manipulation? Could you please help me understand how could I study asymptotic behaviour of $$(4)$$?

$$p(x)=\int_c^x \exp\bigg\{-2\int_c^{\xi}\frac{b(\zeta)}{\sigma^2(\zeta)}d\zeta\bigg\}d\xi$$ which solves the ode $$y'=-\frac{2(c_{1}+c_{2}x)}{x(1-x)}y(x)$$ and so $$p(x) = c x^{-2 c_{1}} (1 - x)^{2 (c_{1} +c_{2})}.$$
So we see that if $$c_{1}>0, c_{1}+c_{2}<0$$, then $$p(0)=p(1)=\infty$$ and so according to 5.22, the process is trapped in $$(0,1)$$.