Question

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

Answer 1

`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`