Integrating hyperbolic functions

In summary, the conversation discusses the integration of the function (tanh(x)+coth(x))/((cosh(x))^2) and the substitution of u=tanhx, leading to an incorrect answer. It is suggested to use the identity (tanhx)^2+(sechx)^2=1 to find the correct solution.
  • #1
peripatein
880
0
Hi,
I am trying to integrate (tanh(x)+coth(x))/((cosh(x))^2)
I am substituting u=tanh(x), du=dx/((cosh(x))^2)
and end up with 1/2(tanh(x))^2 + ln |tanh(x)| + C
which is incorrect. What am I doing wrong??
 
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  • #2
Does someone have an idea what is stymying my answer?
 
  • #3
Why did you chose u = tanh(x)? What happens if you expand (tanh(x) + coth (x))?
 
  • #4
I used u=tanhx, as 1/(coshx)^2 is its derivative.
 
  • #5
peripatein said:
Does someone have an idea what is stymying my answer?

There is nothing wrong with your answer. If the book is giving one that looks different it may differ from yours by a constant.
 
  • #6
Online calculators claim the integral to be -1/2*(coshx)^2 + ln |tanhx| + c.
1/2*(tanhx)^2 (which is the first term in my answer) is not equal to -1/2*(coshx)^2, is it?
 
  • #7
peripatein said:
Online calculators claim the integral to be -1/2*(coshx)^2 + ln |tanhx| + c.
1/2*(tanhx)^2 (which is the first term in my answer) is not equal to -1/2*(coshx)^2, is it?

Wolfram Alpha gives the first term as -(sechx)^2/2 and that does differ from (tanhx)^2/2 by a constant. What's the constant? Are you sure the online calculator isn't saying -1/(2*(coshx)^2)? You should use more parentheses when you write something like -1/2*(coshx)^2. It's ambiguous.
 
Last edited:
  • #8
I am not following your argument. Is the answer which Wolfram's calculator yields equal to mine?
My answer is: (0.5)(tanh(x))^2 + ln |tanh(x)| + C
Wolfram's calculator's answer: (-0.5)(sech(x)^2) + ln [tanh(x)] + C
 
  • #9
peripatein said:
I am not following your argument. Is the answer which Wolfram's calculator yields equal to mine?
My answer is: (0.5)(tanh(x))^2 + ln |tanh(x)| + C
Wolfram's calculator's answer: (-0.5)(sech(x)^2) + ln [tanh(x)] + C

They are only 'equal' if you consider the '+C' part. (tanhx)^2+(sechx)^2=1. Use that identity.
 
  • #10
I see. Thanks a lot!
 

1. What are hyperbolic functions?

Hyperbolic functions are mathematical functions that are related to the hyperbola, a type of curved line in geometry. They are used to model various phenomena in physics, engineering, and other fields.

2. How are hyperbolic functions different from trigonometric functions?

Hyperbolic functions are based on the hyperbola, while trigonometric functions are based on the circle. They have different shapes and properties, and their values are calculated using different formulas.

3. What is the purpose of integrating hyperbolic functions?

Integrating hyperbolic functions allows us to find the area under their curves, which is useful in solving a variety of mathematical problems. It also has practical applications in physics and engineering.

4. What are the common hyperbolic functions?

The most frequently used hyperbolic functions are sineh (sinh), cosineh (cosh), and tangenth (tanh). Other hyperbolic functions include cosech (cosech), sech (sech), and cotanh (coth).

5. What are some real-world applications of integrating hyperbolic functions?

Integrating hyperbolic functions is used in fields such as physics, engineering, and economics to model and solve problems related to heat transfer, electrical circuits, and population growth, among others. It is also used in statistics and probability to calculate probabilities and areas under curves.

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