`Ifint_(-pi/4)^((3pi)/4)((e^(pi/4)dx)/((e^x+e^(pi/4))(sinx+cosx))=kint_(-pi/2)^(pi/2)secx dx ,t h e nt h ev a l u eofki s` `1/2` (b) `1/(sqrt(2))` (c) `1/(2sqrt(2))` (d) `-1/(sqrt(2))`

Options

A. `1/2`
B. `1/(sqrt(2))`
C. `1/(2sqrt(2))`
D. `-1/(sqrt(2))`

Answer

C

Explanation

`I=int_(-pi//4)^(3pi/4)(dx)/(sqrt(2)(e^(x-pi//4)+1)cos(x-(pi)/4))`
Putting `x-(pi)/4=t`, we get
`I=1/(sqrt(2))int_(-pi//2)^(pi//2) (dt)/((e^(t)+1)cost)`…………..1
`=1/(sqrt(2))int_(-pi//2)^(pi//2)(e^(t)dt)/((e^(t)+1)cost)`……………..2
Adding 1 and 2 we get `2I=1/(sqrt(2)) int_(-pi//2)^(pi//2) sec t dt`
`=I=1/(2sqrt(2))int_(-pi//2)^(pi//2) sec x dx`
`:. k=1/(2sqrt(2))`