Does the following improper integral converge ?
$$\int _0^{\frac{1}{2}}\:\cfrac{1}{\sin\left(x\right)\ln\left(x\right)}dx$$
I have tried to compare it to some known improper integrals but with no luck.
Thanks for helping.
2026-04-09 02:03:11.1775700191
Improper integral - check convergence
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Assume $0<\varepsilon<\dfrac12$. From the inequality $$ 0<\sin x < x, \qquad \varepsilon<x<\frac12, $$ one gets $$ \int_\varepsilon^{1/2}\frac{-1}{\sin x \ln x}dx\ge\int_\varepsilon^{1/2}\frac{-1}{x\ln x}dx=\left[-\ln(-\ln x)\frac{}{}\right]_\varepsilon^{1/2}=\ln(-\ln \varepsilon)-\ln(\ln 2) $$ yielding the divergence of the given integral as $\varepsilon \to 0^+$.