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1+cosx = 2[cos(x/2)]^2
∫(0->π/2) x/(1+cosx ) dx
=(1/2)∫(0->π/2) x[sec(x/2)]^2 dx
=∫(0->π/2) x dtan(x/2)
=[xtan(x/2)]|(0->π/2) - ∫(0->π/2) tan(x/2) dx
=π/4 - ∫(0->π/2) sin(x/2)/cos(x/2) dx
=π/4 +2 ∫(0->π/2) dcos(x/2)/cos(x/2)
=π/4 +2 [ln|cos(x/2)|]|(0->π/2)
=π/4 +2ln(1/√2)
=π/4 -ln2
∫(0->π/2) x/(1+cosx ) dx
=(1/2)∫(0->π/2) x[sec(x/2)]^2 dx
=∫(0->π/2) x dtan(x/2)
=[xtan(x/2)]|(0->π/2) - ∫(0->π/2) tan(x/2) dx
=π/4 - ∫(0->π/2) sin(x/2)/cos(x/2) dx
=π/4 +2 ∫(0->π/2) dcos(x/2)/cos(x/2)
=π/4 +2 [ln|cos(x/2)|]|(0->π/2)
=π/4 +2ln(1/√2)
=π/4 -ln2
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