Đặt: $x=\sin t \Rightarrow dx=\cos tdt$
Đổi cận: $x=0 \Rightarrow t=0$
$x=1 \Rightarrow t=\dfrac{\pi}{2}$
Suy ra:
$I=\int\limits_0^{\frac{\pi}{2}}\sin^2t.\cos t\cos t.dt$
$=\frac14\int\limits_0^{\frac{\pi}{2}}\sin^22tdt$
$=\int\limits_0^{\frac{\pi}{2}}\dfrac{1-\cos4t}{8}dt$
$=\left(\dfrac{t}{8}-\dfrac{\sin4t}{32}\right)\left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.=\dfrac{\pi}{16}.$