積分:(2-x^2)^(3/2)dx
=x(2-x^2)^(3/2)-積分:xd(2-x^2)^(3/2)
=x(2-x^2)^(3/2)-積分:x(-2x)*3/2*(2-x^2)^(1/2)dx
=x(2-x^2)^(3/2)+3積分:x^2(2-x^2)^(1/2)dx
現(xiàn)在求:積分:x^2(2-x^2)^(1/2)dx
令:x=根號(2)*cost,t[0,pi/2]
t=arccos(根號(2)*x/2)
所以:
積分:x^2(2-x^2)^(1/2)dx
=積分:2(cost)^2*根號(2)sint*根號(2)*(-sint)dt
=-4積分:(sintcost)^2dt
=-積分:(sin(2t))^2dt
=積分:(cos(4t)-1)dt
=1/4sin4t-t+C
x[0,1]
t[pi/2,pi/4]
該定積分為:
pi/4
積分上下限是:(0,1)
定積分為:
=x(2-x^2)^(3/2)+3積分:x^2(2-x^2)^(1/2)dx
=x(2-x^2)^(3/2)|(0,1)+3*pi/4
=1+3pi/4
因為被積函數(shù)是偶函數(shù),所以最后的定積分為:
I =2*(1+3pi/4)
=2+3pi/3