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

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

A

`(1)/(sqrt(2))tan^(-1)(x-(1)/(x))+C`

B

`(1)/(sqrt(2))cot^(-1){(x-(1)/(x))}+C`

C

`(1)/(sqrt(2))tan^(-1){(1)/(sqrt(2))(x-(1)/(x))}+C`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
On dividing the numerator and denominator by `x^(2)`, we get
`I=int((1+(1)/(x^(2))))/((x^(2)+(1)/(x^(2))))dx=int((1+(1)/(x^(2))))/({(x-(1)/(x))^(2)+2})dx`
`=int(dt)/((t^(2)+2))," where "(x-(1)/(x))=t and(1+(1)/(x^(2)))dx=dt`
`int(dt)/([t^(2)+(sqrt(2))^(2)])=(1)/(sqrt(2))+C=(1)/(sqrt(2))tan^(-1){(1)/(sqrt(2))(x-(1)/(x))}+C`.
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