∫(pai/2到0)sinx/8+sin^2x dx

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∫(pai/2到0)sinx/8+sin^2x dx

∫(pai/2到0)sinx/8+sin^2x dx
∫(pai/2到0)sinx/8+sin^2x dx

∫(pai/2到0)sinx/8+sin^2x dx
∫(π/2到0)[sinx/(8+sin²x)] dx
=-∫(0,π/2)dcosx/(9-cos²x)
=-∫(0,π/2)dcosx/(3+cosx)(3-cosx)
=(-1/6)∫(0,π/2)[1/(3+cosx)+1/(3-cosx)]dcosx
=(-1/6)[ln(3+cosx)-ln(3-cosx)]|(0,π/2)
=(-1/6)×[(ln3-ln3)-(ln4-ln2)]
=(1/6)×ln2
=(ln2)/6

原式=∫(pai/2到0)sinx/8 dx+∫(pai/2到0)(1-cos2x)/2 dx
={(-cosx/8+0.5x-(sin2x/4)}(pai/2到0)
=(-1/8)-(pai/4)