Question

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 `lim_(x to oo) tan^(-1)sqrt(f(x))` is
A. `pi//2`
B. `pi//4`
C. `pi`
D. `2pi`

Answer 1

Correct Answer - A
`int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))dx`
`=int(x(x-1))/((x^(2)+1)(x+1)xsqrt(x+1+(1)/(x)))dx`
`=int(x^(2)-1)/((x^(2)+1)(x+1)^(2)sqrt(x+1+(1)/(x)))dx`
`=int(1-(1)/(x^(2)))/((x+(1)/(x))(x+2+(1)/(x))sqrt(x+1+(1)/(x)))dx`
Put `x+(1)/(x)+1=t^(2)`
` :. I=int(2tdt)/((t^(2)-1)(t^(2)+1)t)`
`=2int(dt)/((t^(2)-1)(t^(2)+1))`
`=int((1)/(t^(2)-1)-(1)/(t^(2)+1))dt`
`=(1)/(2) log|(t-1)/(t+1)|-tan^(-1)t+C`
`=(1)/(2)log_(e)|(sqrt(x+(1)/(x)+1)-1)/(sqrt(x+(1)/(x)+1)+1)|-tan^(-1)sqrt(x+(1)/(x)+1)+C`