`int(dx)/(xsqrt(x^(6)-16))=`

Options

A. `(1)/(3)sec^(-1)((x^(3))/(4))+c`
B. `cos^(-1)((x^(3))/(4))+c`
C. `(1)/(12)sec^(-1)((x^(3))/(4))+c`
D. `sec^(-1)((x^(3))/(4))+c`

Answer

C

Explanation

let `I=int(dx)/(xsqrt(x^(6)-16))`
`=(1)/(3)int(3x^(2))/(x^(3)sqrt((x^(3))^(2)-4^(2)))dx`
Put `x^(3)=t`
`implies3x^(3)dx=dt`
`thereforel=(1)/(3)int(dt)/(tsqrt(t^(2)-4^(2)))`
`=(1)/(3xx4)sec^(-1)((t)/(4))+c`
`=(1)/(12)sec^(-1)((x^(3))/(4))+c`