`T h ev a l u eoft h ein t egr a lint_(-(3pi)/4)^((5pi)/4)((sinx+cosx)/(e^(x-pi/4)+1)dx` 0 (b) 1 (c) 2 (d) none of these

Options

A. `0`
B. `1`
C. `2`
D. none of these

Answer

A

Explanation

Let `I=int_(-3pi//4)^(5pi//4)((sinx+cosx)/(e^(x-pi//4)+1)) dx`
`=int_(-3pi//4)^(5pi//4) (sqrt(2)cos(x-(pi)/4))/(e^(x-pi//4)+1)dx`
Putting `x-(pi)/4t` or `dx=dt`
`:. I=int_(-pi)^(pi)(sqrt(2)cost)/(e^(t)+1)dt`……………1
Replacing `t` by `pi+(-pi)-t` or `-t` we get
`I=int_(-pi)^(pi)(sqrt(2)cos(-t))/(e^(-t)+1)dt=int_(-pi)^(pi)(e^(t)sqrt(2)cost)/(e^(t)+1) dt`.................2
Adding 1 and 2 we get
`2I=sqrt(2)int_(-pi)^(pi)cos tdt `or `I=0`