(36-x^2)^(1/2) 是哪一个的导数
3个回答
(36-x^2)^(1/2)是哪一个的导数,就相当于求(36-x^2)^(1/2)的积分.
∫(36-x^2)^(1/2) dx
= 6∫[1-(x/6)^2]^(1/2) dx
= 36∫[1-(x/6)^2]^(1/2) d(x/6)
设x/6 = sint,则d(x/6) = costdt,[1-(x/6)^2]^(1/2) = [1-(sint)^2]^(1/2) = cost得
36∫[1-(x/6)^2]^(1/2) d(x/6)
= 36∫(cost)^2 dt
= 18∫(1+cos2t) dt
= 18[∫dt + 0.5∫cos2td(2t)]
= 18(t + 0.5sin2t) + C
= 18(t + sintcost) + C（C为任意常数）
由于sint = x/6,因此t = arcsin(x/6),cost = [1-(x/6)^2]^(1/2) = (36-x^2)^(1/2)/2
得18(t + sintcost) + C
= 18[arcsin(x/6) + x(36-x^2)^(1/2)/36] + C
= 18arcsin(x/6) + x(36-x^2)^(1/2)/2 + C
即f(x) = 18arcsin(x/6) + x(36-x^2)^(1/2)/2 + C 的导数f'(x)为(36-x^2)^(1/2),其中C为任意常数.
