integration

As per Nishant's Sir hint :-

2 f(x) + f(-x) = 1/x sin (x - 1/x) ...................................(i)

Replace x by -x , we get

2 f (-x) + f(x) = 1/x sin (x- 1/x) ...................................(ii)

Subtracting (i) and (ii), we get
f(x) = f(-x)

Therefore, 3 f(x) = 1/x sin (x- 1/x)
Therefore, I =
= 1/3
Put 1/x = t , dx = -1/t² dt

Therefore, I = 1/3
= -1/3

Therefore, I = - I
2 I = 0
I = 0