How do you integrate #int (4x)/sqrt(x^2-14x+65)dx# using trigonometric substitution?

1 Answer
Mar 8, 2016

#4(sqrt(x^2-14x+65) +7sinh^-1((x-7)/4))+ C#

Explanation:

The two main tricks are:

  1. See that: #x^2-14x+65 = (x-7)^2 +4^2#
  2. Use and abuse: #sinh^2u + 1 = cosh^2 u#

Using 1. we choose our substitution like this:

  • #x-7 = 4 sinhu#
  • #dx = 4 cosh u*du#

To get the following:

#(x-7)^2 +4^2 -> 4^2sinh^2u + 4^2= 16cosh^2u#

Which will simplify #sqrt(x^2-14x+65) -> 4cosh u#

We can proceed:
#int (4x)/sqrt(x^2-14x+65)dx = int (4*(4sinh u +7))/(4cosh u)(4 cosh u) du#
#= 4int (4sinh u +7)du#
#= 4*(4cosh u +7u)+ c#

We can then substitute it back in x:

#4(sqrt(x^2-14x+65) +7sinh^-1((x-7)/4))+ C#