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

1 Answer
trosk
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#

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