If `sqrt(x+sqrt(x^(2)+x))dx=A{x+sqrt(x^(2)+2)}^(3//2)+(B)/(sqrt(x+sqrt(x^(2)+2)))+c.` then the value of 3AB is

Options

A. `-1`
B. `-2`
C. 1
D. 2

Answer

B

Explanation

Put `x+sqrt(x^(2)+2)=t`
`rArr" "x^(2)+2=t^(2)+x^(2)-2tx`
`rArr" "x=(1)/(2)(t-(2)/(t))`
`"So "I=intsqrt(x+sqrt(x^(2)+2))dx`
`=(1)/(2)intt^(1//2)(1+(2)/(t^(2)))dt`
`=(1)/(2)intt^(1//2)dt+intt^(-3//2)dt=(1)/(3)t^(3//2)-(2)/(sqrtt)+c`