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int(x^(2)+1)/(x^(4)-x^(2)+1)dx= . . ....

`int(x^(2)+1)/(x^(4)-x^(2)+1)dx=` . . .

A

`tan^(-1)((x^(2) +1)/(2))+c`

B

`tan^(-1)(x^(2)) + c`

C

`tan^(-1)(2x^(2) -1) + c`

D

`tan^(-1)((x^(2) -1)/(x))+c`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I = int (x^(2) + 1)/(x^(4) - x^(2) + 1) dx`
`= int (1 + (1)/(x^(2)))/(x^(2) - 1 + (1)/(x^(2))) dx = int (1 + (1)/(x^(2)))/((x - (1)/(x))^(2) + 1)`
Put `x - (1)/(x) = t`
`(1 + (1)/(x^(2))) dx = dt`
`:. I = int (dt)/(t^(2) + 1) = tan^(-1) (t) + c`
`= tan^(-1) (x - (1)/(x)) + C tan^(-1) ((x^(2) - 1)/(x)) + C`
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