1. ## Hyperbolic substitution

Hi,
I am trying some revision for my maths test and I have to use hyperbolic functions to evaluate this integral.

$\displaystyle \int {\frac{3x}{\sqrt{{x^4-9}}}dx }$

I am having trouble with the substitution. I tried letting

$\displaystyle x^2 = 3cosh(u)$

But I still cant manage to find a solution. I suspect something nice is spose to happen with the 3x on the top of the integral but not sure how. Any help would be greatly appreciated.

Thanks
Elbarto

2. Originally Posted by elbarto
Hi,
I am trying some revision for my maths test and I have to use hyperbolic functions to evaluate this integral.

$\displaystyle \int {\frac{3x}{\sqrt{{x^4-9}}}dx }$

I am having trouble with the substitution. I tried letting
$\displaystyle x^2 = 3cosh(u)$
But I still cant manage to find a solution. I suspect something nice is spose to happen with the 3x on the top of the integral but not sure how. Any help would be greatly appreciated.

Thanks
Elbarto
$\displaystyle x^2 = 3 \cosh u \Rightarrow 2x = 3 \sinh u \, \frac{du}{dx} \Rightarrow dx = \frac{3 \sinh u}{2x} \, du$.

3. Thank you for the fast response but I am confused as to why you took the derivative of both sides of the expression. Is this the short hand method of taking the arcosh() of the function or something similar?

4. Originally Posted by elbarto
Thank you for the fast response but I am confused as to why you took the derivative of both sides of the expression. Is this the short hand method of taking the arcosh() of the function or something similar?
I used implicit differentiation to get the derivative. The derivative is needed because when you do the substitution you have to substitute for dx as well.

Note the edit to my first post. I forgot the du in the expression for dx.