求积分∫[arcsin√x/√(1-x)]dx

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求积分∫[arcsin√x/√(1-x)]dx求积分∫[arcsin√x/√(1-x)]dx求积分∫[arcsin√x/√(1-x)]dx令√x=u,则:x=u^2,dx=2udu.∴∫[arcsin

求积分∫[arcsin√x/√(1-x)]dx
求积分∫[arcsin√x/√(1-x)]dx

求积分∫[arcsin√x/√(1-x)]dx
令√x=u,则:x=u^2,dx=2udu.
∴∫[arcsin√x/√(1-x)]dx
=∫[arcsinu/√(1-u^2)]2udu
=-2∫arcsinu{-2u/[2√(1-u^2)]}du
=-2∫arcsinud[√(1-u^2)]
=-2[√(1-u^2)]arcsinu+2∫[√(1-u^2)]d(arcsinu)
=-2[√(1-u^2)]arcsinu+2∫[√(1-u^2)][1/√(1-u^2)]du
=-2[√(1-u^2)]arcsinu+2∫du
=2u-2[√(1-u^2)]arcsinu+C
=2√x-2[√(1-x)]arcsin√x+C

令arcsin√x=t,则x=sin^2(t)
原式=∫t/cost*2sintcostdt
=∫2tsintdt
=-2∫td(cost)
=-2tcost+2∫costdt
=-2tcost+2sint+C
=-2√(1-x)*arcsin√x+2√x+C