Finding Indefinite Integral of a combination of hyperbolic functions

In summary, the integral \int \frac{cosh(x)}{cosh^2(x) - 1}\,dx can be simplified to \int coth(x)csch(x)\,dx by using the identity cosh^2(x) - 1 = sinh^2(x). This can then be solved using the identity \int cosh(x)\,dx = sinh(x) + C, resulting in a final solution of -csch(x) + C.
  • #1
tainted
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Homework Statement



Compute the following:

[itex] \int \frac{cosh(x)}{cosh^2(x) - 1}\,dx [/itex]


Homework Equations


[itex] \int cosh(x)\,dx = sinh(x) + C [/itex]


The Attempt at a Solution


I had no clue where to start, so I went to WolfRamAlpha, and it used substitution but it made [itex] u = tanh(\frac{x}{2}) [/itex] and I had no clue how I am supposed to know to do that.

Any advice on where to start is greatly appreciated.
 
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  • #2
I solved this problem after realizing that [itex] cosh^2(x) - 1 = sinh^2(x) [/itex]. This allowe me to split make it [itex] \int coth(x)csch(x)\,dx [/itex], and then I could use another identity to set that equal to [itex] -csch(x) + c [/itex]
 

1. What are hyperbolic functions?

Hyperbolic functions are mathematical functions that are related to the hyperbola, a type of curve in geometry. Examples of hyperbolic functions include hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh).

2. What is an indefinite integral?

An indefinite integral is the process of finding the original function from its derivative. It is represented by the symbol ∫ (integral sign) and is the inverse operation of differentiation.

3. How do you find the indefinite integral of a combination of hyperbolic functions?

To find the indefinite integral of a combination of hyperbolic functions, you can use the properties of hyperbolic functions and integration techniques such as substitution and integration by parts. It is important to follow the rules of integration and simplify the expression as much as possible.

4. What is the purpose of finding the indefinite integral of a combination of hyperbolic functions?

The purpose of finding the indefinite integral of a combination of hyperbolic functions is to solve mathematical problems that involve these types of functions. This process is also useful in physics and engineering applications, as hyperbolic functions often arise in the modeling of real-world phenomena.

5. Are there any limitations to finding the indefinite integral of a combination of hyperbolic functions?

Yes, there are some limitations when finding the indefinite integral of a combination of hyperbolic functions. For example, if the combination involves complex numbers, the integral may not exist. Additionally, some combinations may require advanced techniques or cannot be solved analytically, in which case numerical methods may be used.

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