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Evaluate: int(x^2+1)/(x^4-x^2+1)\ dx...

Evaluate: `int(x^2+1)/(x^4-x^2+1)\ dx`

Text Solution

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`I=int(x^2+1)/(x^4-x^2+1)\ dx`
`=int frac{1+frac{1}{x^2}}{x^2+frac{1}{x^2}-1}dx`
`=int frac{(1+1/(x^2))}{(x-1/x)^2+1}dx`
Let `x-1/x=t` and `(1+1/(x^2))dx=dt`
`:. I=int frac{dt}{t^2+1}=tan^(-1)t+C`
`=tan^(-1)(x-1/x)+C`
`=tan^(-1)(frac{x^2-1}{x})+C`
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