# Integrate cosh(x)^2

• Jul 15th 2010, 09:30 AM
MechEng
Integrate cosh(x)^2
Good afternoon,

I am working on my last surface area problem, and am kind of hung up on the following:

$\displaystyle \int{cosh(x)^2} dx$

Wait... could that be re-written as?:

$\displaystyle \int{1+sinh(x)^2} dx$

That doesn't really help either...
• Jul 15th 2010, 09:39 AM
Ted
Is it $\displaystyle cosh(x^2)$ ?
If so, this integral is unelementary.

Do you mean $\displaystyle cosh^2(x)$ ?
• Jul 15th 2010, 09:40 AM
chisigma
The easiest way is the following identity...

$\displaystyle \displaystyle \cosh^{2} x = \frac{1}{2} + \frac{1}{2}\ \cosh 2x$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Jul 15th 2010, 09:44 AM
MechEng
It is in fact $\displaystyle cosh^2(x)$ which I have written as $\displaystyle cosh(x)^2$

Where is the identity derived from?
• Jul 15th 2010, 09:45 AM
Ted
Its a well-known identity.
You can prove it by using the definitions of the Heperbolic Functions.

******Edit:******

Another way to do it:

$\displaystyle cosh^2(x) = \left( \dfrac{e^{x}+e^{-x}}{2} \right)^2=\dfrac{e^{2x}+e^{-2x}+2}{2}=\frac{1}{2}e^{2x}+\frac{1}{2}e^{-2x}+1$

and it is easy to integrate.
• Jul 15th 2010, 10:15 AM
MechEng
Would that be the double angle formula for $\displaystyle cos^{2}x=\frac{1+cos2x}{2}$

Pardon my ignorance, but are all of the trig identities applicable to the hyperbolic functions?
• Jul 15th 2010, 10:23 AM
Ted
No. Not all of them.
Also, sometimes there is a small difference like the signs.

for example:

$\displaystyle cos(x+y)=cos(x)cos(y) - sin(x)sin(y)$

but

$\displaystyle cosh(x+y)=cosh(x)cosh(y) + sinh(x) sinh(y)$
• Jul 15th 2010, 10:46 AM
MechEng
Just to check my work...

What is the surface area of $\displaystyle f(x)=\frac{e^{x}+e^{-x}}{2}$ for $\displaystyle 0 \leq x \leq 2$ when revolved about the x-axis?

I get:

$\displaystyle S=\pi(2+sinh(4))$

Is this a reasonable looking answer?
• Jul 15th 2010, 12:07 PM
roninpro
Quote:

Originally Posted by MechEng
Would that be the double angle formula for $\displaystyle cos^{2}x=\frac{1+cos2x}{2}$

Pardon my ignorance, but are all of the trig identities applicable to the hyperbolic functions?

Actually, yes, with some slight modifications. Have a look at Hyperbolic function - Wikipedia, the free encyclopedia, and look for "Osborn's Rule".