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

Options

A. `sec^(-1)((x^(2)+1)/(sqrt(2)x))+c`
B. `(1)/(sqrt(2))sec^(-1)((x^(2+1))/(sqrt(2)x))+c`
C. `(1)/(sqrt(2))sec^(-1)((x^(2)+1)/(sqrt(2)))+c`
D. None of these

Answer

B

Explanation

`int(x^(2)(1-(1)/(x^(2))))/(x^(2)(1+(1)/(x^(2)))xsqrt(x^(2)+(1)/(x^(2))))dx`
`=int((1-(1)/(x^(2))))/((x+(1)/(x))sqrt((x+(1)/(x))^(2)-2))dx`
`x+(1)/(x)=t`
`(1-(1)/(x^(2)))dx=dt`
`thereforeint(dt)/(tsqrt(t^(2)-2))`
`int(dt)/(tsqrt(t^(2)-(sqrt(2))^(2)))`
`rArr(1)/(sqrt(2))sec^(-1)""(t)/(sqrt(2))+c`