`int_(0)^(pi) (x tan x dx)/(sec x +tan x)` का मान ज्ञात कीजिए।

Explanation

मानलो `I=int_(0)^(pi) (x tan x)/(sec x + tan x) dx` ...(i)
`=int_(0)^(pi) ((pi-x) tan (pi-x))/(sec (pi-x)+tan (pi-x)) dx`
`=int_(0)^(pi) ((pi-x) tan x)/(sec x+tan x)` ...(ii)
समीकरण (i) तथा (ii) को जोड़ने पर,
`2I=int_(0)^(pi) (x tan x+(pi-x) tan x)/(sec x +tan x) dx`
`implies 2I=pi int_(0)^(pi) (tan x)/(sec x+ tan x) dx`
`I=pi/2 int_(0)^(pi) (tan x)/(sec x+tan x) dx`
`=pi/2 int_(0)^(pi) (tan x(sec x-tan x))/((sec x+tan x)(sec x-tan x)) dx`
`=pi/2 int_(0)^(pi) (sec x tan x-tan^(2) x) dx`
`=pi/2 int_(0)^(pi) (sec x tan x-sec^(2) x+1) dx`
`=pi/2 [secx-tan x+x]_(0)^(pi)`
`=pi/2 [(sec pi-tan pi+pi)-(sec theta-tan theta+theta)]`
`=pi/2 [-1+pi-1]=pi(pi/2-1)`