Are the following integrals equal for large $\alpha$:
$$ I_1 =\int_{0}^{y} \exp\left(\, -\alpha \sqrt{x(1-x)}\,\right)\, {\rm d}x $$ $$ I_2 =\int_{0}^{y} \exp\left(\, -\alpha \sqrt{x}\,\right)\, {\rm d}x $$
According to the answers I got from this forum they both equal to:
$$ I \sim_{\alpha \sim \infty} \frac{1}{\alpha^2}- {\frac { \left( \alpha\,y+1 \right) {{\rm e}^{ -\alpha\,y}}}{{\alpha}^{2}}}.$$
However, it doesn't make sense since $\sqrt{x}$ and $\sqrt{x(1-x)}$ are very different and the plot of these functions show the difference. If they be equal then following should be equal which is not clearly equal:
$$\int_{y_1}^{y_1+dy} \exp\left(\, -\alpha \sqrt{x(1-x)}\,\right)\, {\rm d}x = \int_{y_1}^{y_1+dy} \exp\left(\, -\alpha \sqrt{x}\,\right)\, {\rm d}x$$