Could someone explain how to evaluate this horrible integral?

In summary, the integral in question is difficult to evaluate and has been producing imaginary numbers when plugged into Mathematica. The integral is used in computing the period of a particle undergoing periodic motion and should yield a real number. It is possible that the integral has been incorrectly derived. However, a simple substitution can be used to solve the integral, resulting in the solution of \frac{\pi}{2\alpha \sqrt{\beta}}. This does not require an elliptic integral.
  • #1
AxiomOfChoice
533
1
The integral is

[tex]
\int_0^{f(E)} \frac{\cosh \alpha x}{\sqrt{1-\beta \cosh^2 \alpha x}} dx,
[/tex]

where

[tex]
f(E) = \frac{1}{\alpha}\cosh^{-1}(\sqrt{\frac{1}{\beta}}),
[/tex]

[tex]\alpha > 0[/tex], and [tex]0<\beta<1[/tex]. The integral has proven very difficult to evaluate. Every time I plug it into Mathematica with values for [tex]\alpha[/tex] and [tex]\beta[/tex] that satisfy the conditions listed above, I get an imaginary number! But that can't happen because the integral is involved in computing the period of a particle undergoing periodic motion, which is necessarily real.

It is possible that the integral will not evaluate to a real number, in which case I've made a mistake in deriving its form. If that's the case, please let me know. Thanks!
 
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  • #2
I wouldn't dare. Looks like an elliptic integral to me.
 
  • #3
Seems like everything inside the integral is real and measurable, and its being integrated over a real interval, so the answer must be real. I know that there are commands to force Maple to assume certain variables are real, between an interval, etc. Maybe try to find the equivalent commands on Mathematical and try it again?
 
  • #4
Ran this through Maple, the answer is (pi/2)(alpha*sqrt(beta))^-1.

Here is an image:
http://img132.imageshack.us/img132/5376/horribleintegralpd1.png

If you want to actually calculate it, my best guess is maybe to try complex contour integration? That's probably not going to work since the upper limit is finite, but worth a try.
 
Last edited by a moderator:
  • #5
AxiomOfChoice said:
The integral is

[tex]
\int_0^{f(E)} \frac{\cosh \alpha x}{\sqrt{1-\beta \cosh^2 \alpha x}} dx,
[/tex]

where

[tex]
f(E) = \frac{1}{\alpha}\cosh^{-1}(\sqrt{\frac{1}{\beta}}),
[/tex]

[tex]\alpha > 0[/tex], and [tex]0<\beta<1[/tex]. The integral has proven very difficult to evaluate. Every time I plug it into Mathematica with values for [tex]\alpha[/tex] and [tex]\beta[/tex] that satisfy the conditions listed above, I get an imaginary number! But that can't happen because the integral is involved in computing the period of a particle undergoing periodic motion, which is necessarily real.

It is possible that the integral will not evaluate to a real number, in which case I've made a mistake in deriving its form. If that's the case, please let me know. Thanks!

A simple substitution does the trick. Setting
[tex]cosh(\alpha x)=t[/tex]
gives:
[tex]\frac{1}{\alpha \sqrt{\beta}} \int_{1}^{\frac{1}{\sqrt{\beta}}} \frac{t dt} {\sqrt{t^2-1} \sqrt{\frac{1}{\beta}-t^2}}[/tex]
Setting now:
[tex]t^2-1=u^2[/tex]
gives:
[tex]\frac{1}{\alpha \sqrt{\beta}} \int_{0}^{\sqrt{\frac{1}{\beta}-1}} \frac{du} {\sqrt{\frac{1}{\beta}-1-u^2} }[/tex]
giving arcsin as solution and after filling in the limits you get the result:
[tex]\frac{\pi}{2\alpha \sqrt{\beta}}[/tex]

So, no elliptic integral here, HallsofIvy :-)

coomast
 
  • #6
That's some deadly integration skill there.
 

Related to Could someone explain how to evaluate this horrible integral?

1. How do I approach evaluating a difficult integral?

Evaluating integrals can be challenging, especially when they are complex or have no obvious solution. The first step is to identify the type of integral you are dealing with (e.g. trigonometric, exponential, etc.) and then use appropriate techniques or formulas to simplify it. You may also need to use integration by parts or substitution to transform the integral into a more manageable form.

2. What are some common strategies for evaluating integrals?

Some common strategies for evaluating integrals include using basic integration rules and formulas, identifying patterns and symmetries, and using substitution or integration by parts. It is also helpful to break down the integral into smaller, more manageable parts and to look for opportunities to cancel or simplify terms.

3. What should I do if I get stuck while evaluating an integral?

If you get stuck while evaluating an integral, take a step back and review the problem. Make sure you have correctly identified the type of integral and have applied the appropriate techniques. If you are still having trouble, try breaking down the integral into smaller parts or seeking help from a tutor or classmate.

4. How can I check my work when evaluating an integral?

One way to check your work when evaluating an integral is to use a graphing calculator or software to plot the function and compare it to your solution. You can also differentiate your answer to see if it matches the original function. Additionally, you can plug your solution back into the original integral to see if it results in the correct value.

5. Are there any tips for solving particularly difficult integrals?

When dealing with particularly difficult integrals, it is important to be patient and persistent. Break the integral down into smaller parts and try different techniques until you find a solution. It can also be helpful to review past examples and practice problems to gain a better understanding of the techniques and strategies used to evaluate integrals.

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