Найти интегралы:
x=2sint; dx=2costdt
∫ sqrt(4-x^2)dx= ∫ sqrt(4-4sin^2t)*2costdt= ∫ 2cost*2costdt=
=4 ∫cos^2tdt= 2 ∫ (1+cos2t)dt=2t+2*(1/2)sin2t+C=
[blue]sint=x/2 ⇒ t=arcsin(x/2)
cost=sqrt(1-sin^2t)=sqrt(1-(x/2)^2)=sqrt(4-x^2)/2
sin2t=2sint*cost=2*(x/2)*sqrt(4-x^2)/2=x*sqrt(4-x^2)/2[/blue]
=2*arcsin(x/2)+(x*sqrt(4-x^2)/2)+C