题目

定积分的换元法应该怎样用?
比如;x属于【1,4】,求（4-x^2)^(1/2)的dingjifen

提问时间：2020-11-03

答案

∫√(a^2-x^2)dx
=a^2∫√[1-(x/a)^2]d(x/a)
x/a=cosu sinu=√(a^2-x^2)/a sin2u=2sinucosu=2x√(a^2-x^2)/a^2
=a^2∫√[1-(cosu)^2]dcosu
=a^2∫ -(sinu)^2du
=a^2∫[(cos2u-1)/2]du
=a^2(sin2u/2-u/2+C）
=a^2*[x√(a^2-x^2)/a^2-arccos(x/a)/2 +C0]
=x√(a^2-x^2)-(a^2/2)arccos(x/a)+C
a=2
∫[1,2] √(4-x^2)dx
= -1*√3+2*(π/3)