How do you integrate ##int ( (x^2) / (sqrt(4 - x^2)) ) dx##?

Let ##x=2sintheta## so that ##sqrt(4-x^2)=2costheta## and ##dx=2costhetad theta##.

Hence

##intx^2/sqrt(4-x^2)dx##

##=int(4sin^2theta)/(2costheta)(2costhetad theta)##

##=4intsin^2thetad theta##

Since ##cos2theta=1-2sin^2theta## and ##sin2theta=2sinthetacostheta##.

##4intsin^2thetad theta##

##=4int(1-cos2theta)/2d theta##

##=2intd theta-2intcos2thetad theta##

##=2theta-sin2theta+C##

##=2theta-2sinthetacostheta+C##

##=2arcsin(x/2)-1/2xsqrt(4-x^2)+C##

Hence

##intx^2/sqrt(4-x^2)dx=2arcsin(x/2)-1/2xsqrt(4-x^2)+C##