就是积分上限是π/2,下限是-π/2,被积函数是根号下cosx-cos^3x

Question

就是积分上限是π/2,下限是-π/2,被积函数是根号下cosx-cos^3x

Answer 1

∫ [π/2,-π/2] (cosx-cos^3x)^(1/2)dx
=∫ [π/2,-π/2](cosx*sin^2x)^(1/2)dx
=∫ [π/2,0](cosx)^(1/2)sinxdx+∫ [0,-π/2](cosx)^(1/2)(-sinx)dx
=-∫ [π/2,0](cosx)^(1/2)d(cosx)+∫ [0,-π/2](cosx)^(1/2)d(cosx)
= -2/3(cosx)^(3/2) [π/2,0] + 2/3(cosx)^(3/2)[0,-π/2]
=4/3