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evaluate dx/1+x^4
I = 1/1+x4 dx 2I = 2/1+x4 dx divide num & denominator by x2 =2/x2 /(x2+1/x2) dx =[1/x2+1 + 1/x2-1 ]/(x2+1/x2) dx =(1/x2+1)dx/x2+1/x2 + (1/x2-1)dx/(x2+1/x2) =(1+1/x2)dx/{(x-1/x)2 + 2} + -(1-1/x2)dx/{(x+1/x)2 - 2} I II put x-1/x = t in first & x+1/x = u in second & then 2I = dt/t2+2 + -du/u2-2 2I = tan-1(t/21/2) / 21/2 + -log(u-21/2/u+21/2)/81/2 + C now put t , u in above eq...
I = 1/1+x4 dx
2I = 2/1+x4 dx
divide num & denominator by x2
=2/x2 /(x2+1/x2) dx
=[1/x2+1 + 1/x2-1 ]/(x2+1/x2) dx
=(1/x2+1)dx/x2+1/x2 + (1/x2-1)dx/(x2+1/x2)
=(1+1/x2)dx/{(x-1/x)2 + 2} + -(1-1/x2)dx/{(x+1/x)2 - 2}
I II
put x-1/x = t in first & x+1/x = u in second & then
2I = dt/t2+2 + -du/u2-2
2I = tan-1(t/21/2) / 21/2 + -log(u-21/2/u+21/2)/81/2 + C
now put t , u in above eq...
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