求定积分:∫(上标是(π/2),下标是0)|sinx-cosx|dx=

来源:学生作业帮助网 编辑:作业帮 时间:2024/07/15 07:17:44
求定积分:∫(上标是(π/2),下标是0)|sinx-cosx|dx=
xRN0~]J.@C <C w6}z>;q2|q47jWޞ~Sn] v@o"k8uY|v%!$9%!APb3_2=c!HQ?2q@_K5 P#F8PSxY^8Q\P %Hr!@͡i6bg@lYWK}p3ihK{$g{@ (=V>9":PIV` ] -imo?<9

求定积分:∫(上标是(π/2),下标是0)|sinx-cosx|dx=
求定积分:∫(上标是(π/2),下标是0)|sinx-cosx|dx=

求定积分:∫(上标是(π/2),下标是0)|sinx-cosx|dx=
∫(0->π/2)|sinx-cosx|dx
=∫(0->π/4)|sinx-cosx|dx +∫(π/4->π/2)|sinx-cosx|dx
= ∫(0->π/4)(cosx-sinx)dx+∫(π/4->π/2)(sinx-cosx)dx
= ∫(0->π/4)cosxdx- ∫(0->π/4)sinxdx+∫(π/4->π/2)sinxdx-∫(π/4->π/2)cosxdx
=sinx|(0->π/4)+cosx|(0->π/4)-cosx|(π/4->π/2)-sinx|(π/4->π/2)
=(√2/2-0)+(√2/2-1)-(0-√2/2)-(1-√2/2)
=√2/2+√2/2-1+√2/2-1+√2/2
=2√2-2

0≤x≤π/4时,cosx≥sinx,π/4≤x≤π/2时,cosx≤sinx
∫(π/2,0) |sinx-cosx| dx
=∫(π/4,0) (cosx-sinx)dx-∫(π/2,π/4) (cosx-sinx)dx
=∫(π/4,0) cosx dx-∫(π/4,0) sinx dx-∫(π/2,π/4) cosx dx+∫(π/2,π/4) sinx dx
=sinx|(π/4,0) -cosx|(π/4,0)-sinx|(π/2,π/4)-cosx|(π/2,π/4)