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int(0)^( pi)(x)/(1+sin x)dx...

`int_(0)^( pi)(x)/(1+sin x)dx`

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`"Let I"= int_(0)^(pi) (x)/(1+sinx)dx`
`rArr " "I=int_(0)^(pi) (pi-x)/(1+sin (pi-x))dx`
`=int_(0)^(pi)(pi-x)/(1+sin x)dx`
Adding equations (1) and (2)
`2I =int_(0)^(pi)(x+pi-x)/(1+sin x)dx`
`=piint_(0)^(pi)(1)/(1+sinx).(1-sin x)/(1-sin x)dx`
`=piint_(0)^(pi)(1-sinx)/(1-sin^(2)x)dx`
`=pi int_(0)^(pi)(1-sinx)/(cos^(2)x)dx`
`=pi int_(0)^(pi)(sec^(2)x-secx tan x) dx`
`=pi [tan x-sec x]_(0)^(pi)`
`=pi [(tan pi-sec pi)-(tan 0-sec 0)]`
` =pi (1+1) =2pi`
`rArr " "I=pi`
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