Ta có:
$\int\limits_0^2\dfrac{\sin2x}{2\sin^2x+\cos^2x}dx$
$=\int\limits_0^2\dfrac{\sin2xdx}{\sin^2x+1}$
$=\int\limits_0^2\dfrac{d(\sin^2x+1)}{\sin^2x+1}$
$=\ln|\sin^2x+1|\left|\begin{array}{l}2\\0\end{array}\right.$
$=\ln\left(\dfrac{3}{2}-\dfrac{\cos4}{2}\right)$