MHB How to Verify the Complex Integral Equals π/(1+n)?

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The discussion focuses on proving the complex integral $$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$ for $n \ne -1$. Participants explore various mathematical techniques, including contour integration and residue theorem, to validate the equation. The integral's behavior is analyzed over the interval from 0 to 2π, emphasizing the role of the imaginary unit $i$ and the function's periodicity. Key points include the simplification of the integrand and the application of limits to handle the complex components. The discussion ultimately aims to establish a rigorous proof for the stated equality.
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How to prove this integral,

$$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$

$n \ne -1$

$i=\sqrt{-1}$
 
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Tony said:
How to prove this integral,

$$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$

$n \ne -1$

$i=\sqrt{-1}$
Rationalise the fraction: $$\frac{\sin t}{ \sin t+ i\sqrt{n+\cos^2 t}} = \frac{\sin t\bigl( \sin t - i\sqrt{n+\cos^2 t}\bigr)}{1+n}.$$
 
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