Evaluate: `int_0^pi(xtanx)/(secx+tanx)\ dx`

Explanation

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