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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
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