1. ## Hyperbolic functions

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

2. 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?

3. I wrote x as 7cosh(u) and ended up with arccosh(x/2) between 7 and 14, i'm not sure that this is right?

4. 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 $\displaystyle u>1 \implies x>\tfrac{7}{5}$.