Réponse :
Bonsoir,
Explications étape par étape :
[tex]Rappels\\\\\dfrac{d(arcsin(\frac{x}{a}))}{dx}=\dfrac{1}{\sqrt{a^2-x^2}}\\\\\int \sqrt{a^2-x^2}\ dx=\dfrac{x*\sqrt{a^2-x^2}}{2}+\dfrac{a^2}{2}*arcsin(\dfrac{x}{a})\\\\\int arcsin(\dfrac{x}{a})\ dx=x*arcsin(\frac{x}{a})+\sqrt{a^2-x^2}\ (par\ parties)\\[/tex]
[tex]I=\int x*arcsin(\dfrac{x}{a})\ dx\\=x*(x*arcsin(\dfrac{x}{a})+\sqrt{a^2-x^2}) -\int (x*arcsin(\dfrac{x}{a})+\sqrt{a^2-x^2}) \ dx\\\\2*I=x*(x*arcsin(\dfrac{x}{a})+\sqrt{a^2-x^2}) -\int x*\sqrt{a^2-x^2})\ dx\\\\I=\frac{1}{2}*(x^2*arcsin(\dfrac{x}{a})+\dfrac{x}{2}*\sqrt{a^2-x^2}-\dfrac{a^2*arcsin(\dfrac{x}{a})}{4})\\\\[/tex]
[tex]\boxed{I=(\dfrac{x^2}{2}-\dfrac{a^2}{4})*arcsin(\dfrac{x}{a})+\dfrac{x*\sqrt{a^2-x^2}}{4} }[/tex]
