y'=(1-x^2)^(1/2)
dy/dx=(1-x^2)^(1/2)
dy=(1-x^2)^(1/2)dx
两边同时积分
∫dy=∫(1-x^2)^(1/2)dx
y=∫(1-x^2)^(1/2)dx
令t=sinx t=arcsinx
y=∫(1-sin^2t)^(1/2)dsint
=∫costdsint
=∫cos^2tdt
=∫(1+cos(2t))/2dt
=t/2+1/4∫cos(2t)d2t
=t/2+sin(2t)/4+C
=(t+sintcost)/2+C
=(t+sint(1-sin^2t)^0.5)/2+C
代入t=arcsinx
=(arcsinx+sinarcsinx(1-sin^2arcsinx)^0.5)/2+C
=(arcsinx+x(1-x^2)^0.5)/2+C