求定积分∫(π/2,-π/2) 根号cos^x-cos^4x dx

题目

求定积分∫(π/2,-π/2) 根号cos^x-cos^4x dx
∫(π/2,-π/2) 根号cos^2x-cos^4x dx

答案

∫(π/2,-π/2) √(cos^2 x-cos^4 x) dx
=∫(π/2,-π/2) √[cos^2 x(1-cos^2 x)] dx
=∫(π/2,-π/2) √[cos^2 x *sin^2 x] dx
=∫(π/2,-π/2) cosx*sinx dx
=∫(π/2,-π/2) sinx d(sinx)
= (1/2)(sinx)^2 |(π/2,-π/2)
= (1/2)(1-1)
=0