Question

What is `f(x) = int 1/sqrt(x^2+1) dx` if `f(2) = 3`?

Answer 1

`f(x) = sinh^-1 x + 3- sinh^-1 2`

Explanation:

• `x = sinh u`
• `dx = (cosh u) du`

Then

`f(x) = int 1/sqrt(x^2+1)dx = int (cosh u)/sqrt(sinh^2+1) du`
`= int (cosh u)/sqrt(cosh^2 u) du = int du = u + C = sinh^-1 x + C`

With the initial condition:

`f(2) = 3 => sinh^-1 2 + C = 3 => C = 3- sinh^-1 2`

So:
`f(x) = sinh^-1 x + 3- sinh^-1 2`