2f(x)+f(-x)=1/xsin(x-1/x). find ∫f(x)dx in integral limits[e,1/e].

Dear sai

2f(x)+f(-x)=1/xsin(x-1/x) .............1

replace x by -x

2f(-x)+f(x)=1/xsin(x-1/x).............2

2*(1) -(2)

 f(x) = (1/3x) sin(x-1/x)

s0   I =limit [e,1/e]  ∫f(x)dx

         =limit [e,1/e]   ∫(1/3x) sin(x-1/x)  dx ................3

nor let x=1/t

  dx=  -1/t²  dt

 I=limit [1/e,e]   ∫(1/3t) sin(t-1/t)  dx

or I=limit [e,1/e]   -∫(1/3x) sin(x-1/x)  dx ..............4

add 3 and 4

2I =0

I=0

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