Integral with hyperbolic: cosh x

[FelixHelix]

I cannot reach the answer for this integral which is part of a bigger question related to discounting investments. I know what the answer to the integral is and have tried all the substitutions and tricks I know. Any pointer would be great!

∫(1/(1+cosh(x))) = tanh(x) + C

Thanks, Felix

 

[arildno]

Well, does:
$$\frac{d}{dx}tanh(x)=\frac{1}{\cosh^{2}(x)}$$ equal your integrand?

 

[FelixHelix]

No, the integrand is 1/(1+cosh(x))

 

[arildno]

The integral is readily evaluated by using:
$$\int\frac{dx}{1+\cosh(x)}dx=\int\frac{dx}{1+\cosh(x)}\frac{1-\cosh(x)}{1-\cosh(x)}dx=\int\frac{1-\cosh(x)}{-\sinh^{2}(x)}dx$$

 

[arildno]

No, the integrand is 1/(1+cosh(x))

Correct!
So, what must the purported answer be called?

 

[FelixHelix]

I see,thats a good trick i hadnt thought of using. But how would you manipulate the integral you end up with?

 

[FelixHelix]

Correct!
So, what must the purported answer be called?

?? I'm not sure??

 

[arildno]

?? I'm not sure??

Maybe what is called TOTALLY WRONG!

To give you a hint:
The correct answer is tanh(x/2)+C, not tanh(x)+C

 

[FelixHelix]

yes a typo indeed - apologies. What substitution do u use to get there though - i don't see it!

 

[arildno]

First off:
You can easily do the integral the way I indicated.
THEN, find the appropriate hyperbolic identities for half-arguments (very analogous to the more well-known half-angle formulas for the trig functions). The result will simplify to tanh(x/2)

 

[arildno]

We have:
$$\sinh(x)=2\cosh(\frac{x}{2})\sinh(\frac{x}{2}),\cosh(x)=\cosh^{2}(\frac{x}{2})+\sinh^{2}(\frac{x}{2})$$
Note the similarity to the trig identities!
You'll probably also need the fundamental identity:
$$1==\cosh^{2}(\frac{x}{2})-\sinh^{2}(\frac{x}{2})$$

 

[FelixHelix]

Ill take a look in the morning - thanks for your help!