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Evaluate the following integral : int(...

Evaluate the following integral :
`int(1)/(x^(4)+1)dx`

Text Solution

Verified by Experts

We have
`(1)/(x^(4)+1)=(1)/(2)[(x^(2)+1)/(x^(4)+1)-(x^(2)-1)/(x^(4)+1)]`
`=(1)/(2)[(x^(2)+1)/(x^(4)+1)]-(1)/(2)[(x^(2)-1)/(x^(4)+1)]`
`impliesint(1)/(x^(4)+1)dx=(1)/(2)int(x^(2)+1)/(x^(4)+1)dx-(1)/(2)int(x^(2)-1)/(x^(4)+1)dx`
`=(1)/(2sqrt(2))tan^(-1).(x^(2)-1)/(xsqrt(2))-(1)/(4sqrt(2))log|(x^(2)-sqrt(2)x+1)/(x^(2)+sqrt(2)x+1)|+k`
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