I have to find the integral of 1/((25x^2-49)^1/2) using hyperbolic functions. I'm not sure how to start to know which functions to use, thanks
It sounds like you are expected to know (or be able to determine from a table) the derivatives of the inverse hyperbolic functions. In particular, the one that is useful in this problem would be:
$\displaystyle \frac{d}{dx} \cosh^{-1}(x) = \frac{1}{\sqrt{x^2-1}}$
(for $\displaystyle x>1$)
Can you solve this with this information?
You didn't say what the original endpoints were.
I'm not sure which simplification method you use, but this is how I get it:
$\displaystyle \int \frac{1}{\sqrt{25x^2-49}} dx$
$\displaystyle =\frac{1}{7} \int \frac{1}{\sqrt{\tfrac{25}{49}x^2-1}} dx$
$\displaystyle =\frac{1}{7} \int \frac{1}{\sqrt{\left(\tfrac{5}{7}x\right)^2-1}} dx$
Let $\displaystyle u=\tfrac{5}{7}x$ and $\displaystyle du=\tfrac{5}{7}dx$.
$\displaystyle =\frac{1}{5} \int \frac{1}{\sqrt{u^2-1}} du$
$\displaystyle =\tfrac{1}{5} \cosh^{-1} u + C$
$\displaystyle =\tfrac{1}{5} \cosh^{-1}\left(\tfrac{5}{7}x\right) + C$
And this is valid for $\displaystyle u>1 \implies x>\tfrac{7}{5}$.