Đặt x=sint khi đó dx=costdt

Đổi cận: với $x=0\Rightarrow t=0;\text{   }x=\frac{\sqrt{2}}{2}\Rightarrow t=\frac{\pi }{4}$

Ta có:$I=\int\limits_{0}^{\frac{\pi }{4}}{\frac{{{\sin }^{2}}tc\text{os}t}{\sqrt{1-{{\sin }^{2}}t}}}dt=\int\limits_{0}^{\frac{\pi }{4}}{\frac{{{\sin }^{2}}tc\text{os}t}{\sqrt{{{\cos }^{2}}t}}}dt=\frac{1}{2}\int\limits_{0}^{\frac{\pi }{4}}{(1-c\text{os2t})dt=}\left. \frac{1}{2}\left( t-\frac{1}{2}\sin 2t \right) \right|_{0}^{\frac{\pi }{4}}=\frac{\pi }{8}-\frac{1}{4}$