Integrali riassuntivi.

$ \int x^2 arcsin x \, dx = $     Per parti. 

• fattore finito $f(x) = arcsin x \; ⇒\; f'(x) = \frac{1}{\sqrt{1-x^2}} $
• fattore differ. $g'(x) = x^2 \; ⇒ \; g(x) = \frac{x^3}{3} $

$ =\frac{x^3}{3} arcsin x - \frac{1}{3} \int \frac{x^3}{\sqrt{1-x^2}} \, dx = $

Per sostituzione. $ t = 1-x^2 \; ⇒ \; x^2= 1-t \; ⇒ \; 2xdx = - dt \; ⇒ \; xdx = -\frac{dt}{2} $   

$ = \frac{x^3}{3} arcsin x + \frac{1}{6} \int \frac{1-t}{t^{\frac{1}{2}}} \, dt = $

$ = \frac{x^3}{3} arcsin x + \frac{1}{6} \int t^{-\frac{1}{2}}\, dt -  \frac{1}{6} \int t^{\frac{1}{2}} \, dt = $ 

$ = \frac{x^3}{3} arcsin x + \frac{1}{3} t^{\frac{1}{2}} - \frac{1}{9} \int t^{\frac{3}{2}} = $ 

$ = \frac{x^3}{3} arcsin x + \frac{1}{9} \left(3\sqrt{(1-x^2)}-(1-x^2)^{\frac{3}{2}} \right) + c = $ 

$ = \frac{x^3}{3} arcsin x + \frac{1}{9} \left(3\sqrt{(1-x^2)}-(1-x^2)\sqrt{(1-x^2)} \right)+ c = $ 

$ = \frac{x^3}{3} arcsin x + \frac{1}{9} \sqrt{(1-x^2)} (3-(1-x^2))+ c = $

$ = \frac{x^3}{3} arcsin x + \frac{1}{9} \sqrt{(1-x^2)} (2+x^2))+ c $