Question

What is `f(x) = int x/sqrt(x^2-1) dx` if `f(3) = 0`?

Answer 1

`sqrt(x^2 - 1) - sqrt 8`

Explanation:

there's a very obvious pattern here

`f(x) = int dx qquad color(red)(x)/sqrt(color{red}{x^2}-1)`

we know that

`d/dx (y^(1/2)) = 1/2* y^(-1/2) * y'`

so we can trial estimate that if

`F(x) = alpha sqrt(x^2 - 1)`

then

`(dF)/dx = 1/2 alpha 1/sqrt(x^2 - 1) * 2x`

`= alpha \ x/sqrt(x^2 - 1)`

so `alpha = 1`

and

`f(x) = int dx qquad color(red)(x)/sqrt(color{red}{x^2}-1)`

`= sqrt(x^2 - 1) + C`

you can of course play around with substitutions but seeing the pattern means that you can pretty much do the integration in your head. what i am really trying so say is that the integration is very trivial if you see the pattern, otherwise you need to get stuck into a lot of guessing and associated theory

it wold be interesting to see that array of subs that lead to an easy solution. i'd go for the simple `u^2 = x^2 -1` as an opening gambit

so, next, the initial value: `f(3) = 0`

`0 = sqrt 8 + C`

`implies f(x) = sqrt(x^2 - 1) - sqrt 8`