[tex]\displaystyle \\ \\ I_1 = \int\limits_{0}^1 x\sqrt{1-x^2}\, dx = \int\limits_{0}^1 \sqrt{1-x^2}\cdot x\, dx \\ \\ \sqrt{1-x^2} = t \Big|^2 \Rightarrow 1-x^2 = t^2 \Rightarrow -2x\, dx = 2t \, dt \Rufgrarrow \\ \Rightarrow x \, dx = -t\, dt \\ \\ x = 0 \Rightarrow t = \sqrt{1-0} = 1 \\ x = 1 \Rightarrow t = 0 \\ \\ I_1 = \int\limits_{1}^0 t\cdot (-t) \, dt = \int\limits_{0}^1 t\cdot t \, dt = \int\limits_{0}^1t^2 \, dt = \\ \\ = \dfrac{t^3}{3}\Big|_0^1 = \dfrac{1}{3}-\dfrac{0}{3} = \boxed{\dfrac{1}{3}}[/tex]
