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

1 Answer
Jul 2, 2016

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