While computing this integral , I am having trouble at finding residue at -i. My work is like this , but there is mistake in here and I haven't been able to figure it out.
Do you need to use complex integration? First of all, the function is even, so $\displaystyle \begin{align*} \int_{-1}^1{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} &= 2 \int_0^1{ \frac{\mathrm{d}x}{\sqrt{1 - x^2}} } \end{align*}$. Then
$\displaystyle \begin{align*} 2\int_0^1{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} &= 2\lim_{\epsilon \to 1} \int_0^{\epsilon}{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} \\ &= 2\lim_{\epsilon \to 1} \left[ \arcsin{(x)} \right] _0^{\epsilon} \\ &= 2\left[ \lim_{\epsilon \to 1} \arcsin{( \epsilon ) } - \arcsin{(0)} \right] \\ &= 2\arcsin{(1)} \\ &= 2 \left( \frac{\pi}{2} \right) \\ &= \pi \end{align*}$