Consider the definite integral $\displaystyle I = \int_{\pi/6}^{\pi/3} \frac{2x}{\sin 2x}\,dx.$ The factor $x$ in the numerator makes a direct attack hopeless, since $\dfrac{x}{\sin 2x}$ has no antiderivative that can be written down by the usual rules, and integration by parts only makes the integrand worse. Yet the value of $I$ is a short closed expression, with no special functions in it. Look closely at the interval of integration, and at how the denominator behaves on it, before you start any calculation. You may use $\displaystyle\int \frac{2\,dx}{\sin 2x} = \ln|\tan x| + C$. What is the value of $I$?