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

`int(1-x^(2))/((1+x^(2))sqrt(1+x^(4)))dx` is equal to

A

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

B

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

C

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

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
b
Let `I=int(1-x^(2))/((1+x^(2))sqrt(1+x^(4)))dx=int((1)/(x^(2))-1)/((x+(1)/(x))sqrt(x^(2)+(1)/(x^(2))))dx`
`rArrI=-int(1)/((x+(1)/(x))sqrt((x+(1)/(x))^(2)-(sqrt(2))^(2)))d(x+(1)/(x))`
`rArrI=(1)/(sqrt(2))"cosec"^(-1)((x+(1)/(x))/(sqrt(2)))+C=(1)/(sqrt(2))sin^(-1)((sqrt(2)x)/(x^(2)+1))+C`
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