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

1 Answer
Apr 21, 2016

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