If `int(xln(x+sqrt(1+x^2)))/(sqrt(1+x^2))dx =asqrt(1+x^2)ln(x+sqrt(1+x^2))+b x+c` , then (A) `a=1,b =-1` (B) `a=1,b=1` (C) `a=-1, b=1` (D) `a=-1, b=-1`

Options

A. `a=1,b=-1`
B. `a=1,b=1`
C. `a=-1,b=1`
D. `a=-1,b=-1`

Answer

A

Explanation

`I=int x(In(x+sqrt(x^(2)+1)))/(sqrt(x^(2)+1))dx`
`"Let " t=sqrt(x^(2)+1)`
`"or " (dt)/(dx)=(x)/(sqrt(x^(2)+1))`
` :. I=int In(t+sqrt(t^(2)-1))dt `
`=In(t+sqrt(t^(2)-1))t-int(1+(t)/(sqrt(t^(2)-1)))/(t+sqrt(t^(2)-1))tdt`
`=t " In"(t+sqrt(t^(2)-1))-(1)/(2)int(2t)/(sqrt(t^(2)-1))dt `
`=t" In"(t+sqrt(t^(2)-1))-sqrt(t^(2)-1)+C`
`=sqrt(1+x^(2)) In (x+sqrt(1+x^(2)))-x+C`
`"or " a=1,b=-1`