$$ \int_{0}^{2\pi}\frac{1}{1+\cos x}dx $$ I don't know how to do the integration when the singularity is right on the integration path
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The value is $+\infty$. – geetha290krm Jan 10 '23 at 12:34
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Hint: for small $\epsilon>0$,$$\int_{\pi-\epsilon}^\pi\frac{dx}{1+\cos x}=\int_0^\epsilon\frac12\csc^2\frac{x}{2}dx>\frac12\epsilon\csc^2\frac{\epsilon}{2}>\frac{2}{\epsilon}.$$ – J.G. Jan 10 '23 at 12:36
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Are you looking for the Cauchy Principal Value or trying to show the integral diverges? – Accelerator Jan 10 '23 at 13:56