∫(x*arcsin(x)dx
Пусть
u=arcsin(x) du=dx/√(1-x^2)
dv=xdx v=x^2/2
Далее интегрируем по частям
∫(x*arcsin(x)dx=x^2*arcsin(x)/2 -(1/2)*∫(x²dx/√(1-x²)=
x=sin(t)
dx=cos(t)
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)cos(u)du/√(1-sin²(u))=
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)cos(u)du/cos(u))=
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)du=
=x²*arcsin(x)/2 -(1/4)*∫(1-cos(2u)du=
=x²*arcsin(x)/2 -du/4 +(1/4)*∫cos(2u)du=
=x²*arcsin(x)/2 -u/4 +(1/8)*sin(2u)+c=
=x²*arcsin(x)/2 -arcsin(x)/4 +(x*√(1-x²)/4)*sin(2u)+c
∫(x*arcsin(x)dx
Пусть
u=arcsin(x) du=dx/√(1-x^2)
dv=xdx v=x^2/2
Далее интегрируем по частям
∫(x*arcsin(x)dx=x^2*arcsin(x)/2 -(1/2)*∫(x²dx/√(1-x²)=
Пусть
x=sin(t)
dx=cos(t)
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)cos(u)du/√(1-sin²(u))=
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)cos(u)du/cos(u))=
=x²*arcsin(x)/2 -(1/2)*∫(sin²(u)du=
=x²*arcsin(x)/2 -(1/4)*∫(1-cos(2u)du=
=x²*arcsin(x)/2 -du/4 +(1/4)*∫cos(2u)du=
=x²*arcsin(x)/2 -u/4 +(1/8)*sin(2u)+c=
=x²*arcsin(x)/2 -arcsin(x)/4 +(x*√(1-x²)/4)*sin(2u)+c