Integration of hyperbolic functions

In summary, the given problem involves integrating cosh(2x)sinh^2(2x)dx. The solution to this problem involves using the substitution u = sinh(2x) to obtain the integral of sinh^2(2x)2cosh(2x) dx, which simplifies to sinh^3(2x)/6 + C. The confusion regarding the 2cosh(2x) factor is cleared up by the substitution.
  • #1
Agent M27
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Homework Statement



[tex]\int cosh(2x)sinh^{2}(2x)dx[/tex]

Homework Equations



Not sure

The Attempt at a Solution



This was an example problem in the book and was curious how they got to the following answer:

[tex]\int cosh(2x)sinh^{2}(2x)dx = [/tex] [tex]\frac{1}{2}[/tex][tex]\int sinh^{2}(2x)2cosh(2x) dx[/tex]

= [tex]\frac{sinh^{3}2x}{6} + C[/tex]

My issue with this problem is I don't understand what happened to the [tex]2cosh(2x)[/tex]. It relates to [tex]sinh^{2}(x)+cosh^{2}(x)[/tex] but that only equals 1 in normal trig, not hyperbolic. Thanks in advance.

Joe
 
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  • #2
Agent M27 said:

Homework Statement



[tex]\int cosh(2x)sinh^{2}(2x)dx[/tex]

Homework Equations



Not sure
Hyperbolic trig identities would be very relevant.
Agent M27 said:

The Attempt at a Solution



This was an example problem in the book and was curious how they got to the following answer:
For some reason, your LaTeX wasn't showing up correctly. I fixed it by removing several pairs of [ tex] and [ /tex] tags.
Tip: Use one pair of these tags per block.
Agent M27 said:
[tex]\int cosh(2x)sinh^{2}(2x)dx = \frac{1}{2}\int sinh^{2}(2x)2cosh(2x) dx

= \frac{1}{2}\frac{sinh^{3}2x}{3} + C[/tex]

My issue with this problem is I don't understand what happened to the [tex]2cosh(2x)[/tex]. It relates to [tex]sinh^{2}(x)+cosh^{2}(x)[/tex] but that only equals 1 in normal trig, not hyperbolic. Thanks in advance.

Joe

The integration was done using an ordinary substitution, u = sinh(2x).
 
  • #3
Ya I just realized that if I set u= sinh(2x) then du=2cosh(2x) dx then substitute from there. Thanks

Joe
 

1. What are hyperbolic functions?

Hyperbolic functions are mathematical functions that are based on the hyperbolic cosine and hyperbolic sine. They are often used in calculus and physics to model various phenomena, such as the shape of a hanging chain or the motion of a swinging pendulum.

2. How are hyperbolic functions different from trigonometric functions?

Hyperbolic functions are similar to trigonometric functions, but they use an exponential instead of a circular function. This results in a different shape for their graphs and different properties, such as being unbounded and having a more symmetrical behavior.

3. What is the integration of hyperbolic functions used for?

The integration of hyperbolic functions is used to solve various mathematical problems, such as finding the area under a hyperbolic curve or calculating the volume of a hyperbolic solid. It is also used in physics and engineering to model and analyze various systems and phenomena.

4. What are the basic integration rules for hyperbolic functions?

The basic integration rules for hyperbolic functions are similar to those for trigonometric functions. For example, the integral of the hyperbolic cosine is the natural logarithm of the absolute value of its argument, and the integral of the hyperbolic sine is the negative hyperbolic cosine. However, the specific rules may vary depending on the specific hyperbolic function being integrated.

5. Are there any real-world applications of the integration of hyperbolic functions?

Yes, there are many real-world applications of the integration of hyperbolic functions. For example, it is used in finance to model interest rates and in physics to analyze the behavior of springs and elastic materials. It is also used in engineering to design and optimize structures and systems.

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