Is $\int_0^1 \frac{\cos x}{x \sin^2 x}dx$ divergent or convergent? 

Note that 
\[\int_0^1\frac{\cos x}{x \sin^2 x}dx \geq \int_0^{\frac{\pi}{3}}\frac{\cos x}{x \sin^2 x}dx \geq \int_0^{\frac{\pi}{3}}\frac{1}{x \sin^2 x}dx\]
\[\geq \int_0^{\frac{\pi}{3}}\frac{1}{x}dx=\infty.\]
Hence the given integral is divergent.