 Hello and welcome to the session. In this session we discussed the following question which says evaluate integral 0 to pi x tan x upon set x plus tan x dx. Before moving on to the solution let's recall the property of definite integrals which says that integral 0 to a fx dx is equal to integral 0 to a f of a minus x dx. This is the key idea that we use in this question. Let's now move on to the solution. We want to evaluate this integral so we suppose that I be equal to the integral 0 to pi x tan x upon this integral 0 to pi x into, now tan x can be written as tan x upon this one upon set x can be written as 1 upon cos x plus tan x can be written as tan x upon cos x to be equal to integral upon 1 plus the definite integral stated in the key idea integral 0 to a fx dx is equal to integral 0 to a f of a minus x dx. So using this property here we get this is equal to integral 0 to pi in place of x here we put pi minus x so pi minus x into this one upon 1 plus a minus x into integral this is equal to integral 0 to pi dx minus this is equal to minus integral 0 to pi x dx here in place of is equal to pi into integral 0 to pi so from here I is equal to I into integral 0 to pi now the sin x can be written as minus 1 this one upon 1 plus this gives us into integral 0 to pi 1 upon this gives us 2I equal to pi into integral 0 to pi 1 into dx integral 0 to pi 1 upon equal to pi into 2 to pi now how we multiply the numerator and denominator of this integral by 1 minus sin x so we hold into 1 plus sin x you hold 0 to pi this gives us 2I equal to pi square 0 to pi equal to pi square minus pi into integral 0 to pi 1 upon that is we now have 2I equal to pi square 0 to pi into 1 upon cos x which is equal to tan x into this gives us 2I is equal to 0 to pi fx dx minus integral 0 to as we get 2I is equal to pi square 0 to pi integral of integral of sin x into tan x is sin x 0 to pi I is equal to pi square I is equal to pi square 0 minus tan 0 is 0 pi square minus equal to pi square minus 2 pi so from here we get I is equal to pi square upon 2 or you can say we now have I equal to pi upon 2 into 0 to pi x tan x dx is equal to pi by 2 into pi minus 2 the whole so I will answer this session hope you understood the solution of this question