求定积分∫(2-x^2)^-1/2 dx ,上限是根号2,下限是0.
问题描述:
求定积分∫(2-x^2)^-1/2 dx ,上限是根号2,下限是0.
应该是∫(2-x^2)^(1/2) dx
答
I=∫(0,2)(2-x^2)^(1/2) dx
let
x= √2sina
dx=√2cosada
x=0,a=0
x=2,a=π/4
I=∫(0,π/4)2(cosa)^2 da
=∫(0,π/4)(1+cos2a) da
= [a + (sin2a)/2](0,π/4)
= π/4 +1