If `int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))dx`
`=(1)/(2)log_(e)|(sqrt(f(x))-1)/(sqrt(f(x))+1)|-tan^(-1)sqrt(f(x))+C,` then
The value of `underset(x to oo)(lim)tan^(-1)sqrt(f(x))` is

Options

A. `cos^(-1)sqrt(1+(1)/(x))`
B. `tan^(-1)sqrt(x+(1)/(x)+1)`
C. `cot^(-1)sqrt(x+(1)/(x))`
D. `sin^(-1)sqrt(x+(1)/(x)+1)`

Answer

B

Explanation

We have `int(x(x-1))/((x^(2)+)(x+1)sqrt(x^(3)+x^(2)+x))dx`
`=int(x(x^(2)-1))/((x^(2)+1)(x+1)^(2)sqrt(x^(3)+x^(2)+x))dx`
`=int(x^(3)(1-(1)/(x^(2))))/(x^(3)(x+(1)/(x))(sqrtx+(1)/(sqrtx))^(2)sqrt(x+(1)/(x)+1))dx`
`=int((1-(1)/(x^(2))))/((x+(1)/(x))(x+(1)/(x)+2)sqrt(x+(1)/(x)+1))dx`
`I=int(2t)/((t^(2)-1)(t^(2)+1)sqrt(t^(2)))dt`
where `x+(1)/(x)+1=t^(2)`
`=int(2)/((t^(2)-1)(t^(2)+1))dt`
`=int(1)/(t^(2)-1)dt-int(1)/(t^(2)+1)dt`
`=(1)/(2)log|(t-1)/(t+1)|-tan^(-1)t+c`
`=(1)/(2)log|(sqrt(x+(1)/(x)+1)-1)/(sqrt(x+(1)/(x)+1+1))|-tan^(-1)sqrt(x+(1)/(x)+1)+c`