Home
Class 12
MATHS
Evaluate: int0^pi(xtanx)/(secx+tanx)\ d...

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

Text Solution

Verified by Experts

`" 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)`
Promotional Banner

Topper's Solved these Questions

  • INTEGRATION

    NAGEEN PRAKASHAN|Exercise Exercise 7.11|21 Videos

  • DIFFERENTIAL EQUATIONS

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise|18 Videos

  • INVERES TRIGONOMETRIC FUNCTIONS

    NAGEEN PRAKASHAN|Exercise Miscellaneous Exercise (prove That )|9 Videos

Similar Questions

Explore conceptually related problems

Evaluate: int_0^pi(xtanx)/(s e c x+tanx)dx

int(tanx)/(secx+tanx)dx=

int_0^pi tanx/(sinx+tanx)dx

Prove that int_(0)^(pi)(xtanx)/((secx+tanx))dx=pi((pi)/(2)-1) .

Evaluate int(tanx)/(secx+tanx)dx

int_(0)^(pi)(x tanx)/(secx+cosx)dx is

int(1)/(secx+tanx)dx=

intsecx/(secx+tanx)dx

Evaluate the following integral: int_0^pi(xtanx)/(secx\ cos e c\ x)dx