1. ## Hyperbolic function question

Could someone explain how to do this question? I dont fully understand how you could use hyperbolic functions to show the function..

2. Is...

$\displaystyle \frac{x^{4}-1}{x^{4}+1} = \frac{x^{2}-x^{-2}}{x^{2} + x^{-2}} = \frac{e^{2 \ln x} - e^{-2 \ln x}} {e^{2 \ln x} + e^{-2 \ln x}} = \tanh (2 \ln x)$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Originally Posted by Monster32432421
Could someone explain how to do this question? I dont fully understand how you could use hyperbolic functions to show the function..
Is this for the purposes of integration?

$\displaystyle \frac{x^4 - 1}{x^4 + 1} = \frac{x^4 + 1 - 2}{x^4 + 1}$

$\displaystyle = \frac{x^4 + 1}{x^4 + 1} - \frac{2}{x^4 - 1}$

$\displaystyle = 1 - \frac{2}{x^4 - 1}$.

Now use partial fractions on the second term. You will need to use the Partial Fractions method twice.

Once you have done this, when you integrate, you will have one term that involves a logarithm and one term that involves a trigonometric/hyperbolic substitution.

4. ^
nah. they are some practice questions which I wasnt sure of doing.

5. For the exact same question.. how would i turn
$\displaystyle (64x^6+1)/x^3$ in terms of hyperbolic functions, logarithms and other functions..

Would i divide the thing to get 64x^3+ x^-3 then i have no idea how to get the e^xs

6. Is...

$\displaystyle \frac{64 x^{6} + 1}{x^{3}} = 64 x^{3} + x^{-3} = 65 \cdot \cosh (3 \ln x) + 63 \cdot \sinh (3 \ln x)$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

7. Originally Posted by chisigma
Is...

$\displaystyle \frac{64 x^{6} + 1}{x^{3}} = 64 x^{3} + x^{-3} = 65 \cdot \cosh (3 \ln x) + 63 \cdot \sinh (3 \ln x)$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
idk if thats right..... cause the answer is 16cosh(3In(2x)) and my friend got the answer just then..