what is intigration of 1/(1+x^4)

its lower limit is 0 and pper limit is infinity

I = 1/1+x⁴ dx

 2I = [  2/1+x⁴]dx

divide nume and denom by  x²

2I = [ 2/x² /(x² +1/x²)]dx

2I = (1/x²+1) /[x²+1/x²] dx   +  (1/x² -1)/[x²+1/x²]dx

2I                            =          (1+1/x²)/[(x-1/x)²+2] dx                 +       (1/x² -1)/[(x+1/x)²-2] dx

2I                            =                            I₁                                     +                     I₂

 2I                           =              I₁ = 1+1/x² /[(x-1/x)²+2]dx                  +                  I₂ =  (1/x² -1)/[(x+1/x)² -2]dx

 2I                           =                put  here   (x-1/x) = t                               &                      put  (x+1/x) = u

 2I                           =                  I₁ = dt/[t²+2]                                           +                 I₂ = -dt/[u² -2]

2I                            =        I₁ = [tan^-1(t/sqrt2)]/sqrt2                      +                 I₂ = log(u+sqrt2/u-sqrt2)/2sqrt2      +c

2I                           =       I₁ = 1/sqrt2.tan^-1(x²-1/xsqrt2)          +         I₂ = 1/2sqrt2 .log(x²+1+xsqrt2/x²+1-xsqrt2) + c

this is the required result

Approved