Đặt: $x=\sin t \Rightarrow dx=\cos tdt$
Đổi cận: $x=0 \Rightarrow t=0$
$x=\dfrac{\sqrt2}{2} \Rightarrow t=\dfrac{\pi}{4}$
Suy ra:
$I=\int\limits_0^{\frac{\pi}{4}}\dfrac{\sin^2t}{\cos t}\cos tdt$
$=\int\limits_0^{\frac{\pi}{4}}\sin^2tdt$
$=\int\limits_0^{\frac{\pi}{4}}\dfrac{1-\cos2t}{2}dt$
$=\left(\dfrac{t}{2}-\dfrac{\sin2t}{4}\right)\left|\begin{array}{l}\dfrac{\pi}{4}\\0\end{array}\right.=\dfrac{\pi}{8}-\dfrac{1}{4}$