定积分计算 ∫sinx/(1+sinx)dx
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∫(0到π/4) sinx/(1+sinx) dx
=∫(0到π/4) (1+sinx-1)/(1+sinx) dx
=∫(0到π/4) [1-1/(1+sinx)] dx
=∫(0到π/4) dx-∫(0到π/4) (1-sinx)/[(1+sinx)(1-sinx)] dx
=π/4-∫(0到π/4) (1-sinx)/cos²x dx
=π/4-∫(0到π/4) (sec²x-secxtanx) dx
=π/4-[tanx-secx](0到π/4)
=π/4-[(1-√2)-(-1)]
=π/4-2+√2
=∫(0到π/4) (1+sinx-1)/(1+sinx) dx
=∫(0到π/4) [1-1/(1+sinx)] dx
=∫(0到π/4) dx-∫(0到π/4) (1-sinx)/[(1+sinx)(1-sinx)] dx
=π/4-∫(0到π/4) (1-sinx)/cos²x dx
=π/4-∫(0到π/4) (sec²x-secxtanx) dx
=π/4-[tanx-secx](0到π/4)
=π/4-[(1-√2)-(-1)]
=π/4-2+√2
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