`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 `l=int(dx)/(xsqrt(x^(6)-16))=(1)/(3)int(3x^(2))/(x^(3)sqrt((x^(3))^(2)-4^(2)))dx`
Put `x^(3)=t rArr 3x^(2)dx=dt`
`therefore" "l=(1)/(3)int(dt)/(sqrt(t^(2)-4^(2)))=(1)/(3xx4)sec^(-1)((t)/(4))+C`
`=(1)/(12)sec^(-1)((x^(3))/(4))+C`