1. ## Integrals

does the integral $
\int_0^1 {\frac{{e^x }}{{\sqrt {1 - \cos (x)} }}}

$
converge or diverge???

and if you can show how do I see it!
2. Apply limit comparison test with $\int_0^1\frac{dx}{\sqrt x},$ which converges, hence $\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}}{\sqrt{1-\cos x}}\div \frac{1}{\sqrt{x}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{x}{{e}^{x}}}{\sqrt{1-\cos x}}=\underset{x\to 0}{\mathop{\lim }}\,\sqrt{\frac{x}{1-\cos x}}\cdot {{e}^{x}}=0,$ since $\frac{x}{1-\cos x}\to0$ as $x\to0.$ Finally, the integral converges.