Please give any hints about this integral

  • Thread starter bLaf
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In summary, the conversation revolves around finding a solution for the integral of sqrt(x^2-4)/x^4 and various suggestions are given, including using trig substitutions like u=2*sec(x), u=2*sin^-1(x), and x=2cosh(u). Some discussion is also had about the lack of teaching of secant and cosecant functions in certain countries.
  • #1
bLaf
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Hi guys, I and my colleague have been struggling with this integral for over 2 hours. :D

Any hints about solving it are welcome.

[tex]\int \frac{\sqrt{x^2 - 4}}{x^4}[/tex]


Thanks in advance :)
 
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  • #2
Looks like a trig substitution to me. Try u=2*sec(x).
 
  • #3
Hi Dick,

thanks for your reply.

Unfortunately, in some countries(like mine) the secant and cosecant trig functions are not taught(not even in mathematical schools).

Aren't there any alternative approaches? Isn't there a way to use sin/cos/tg/cotg substitutions?

Thanks!
 
  • #4
You could try [tex]u=2*sin^{-1}(x)[/tex]
 
  • #5
The simplest is to use the hyperbolic cosine substitution, x=2Cosh(u).

Using this, you'll readily find a proper anti-derivative, namely:
[tex]A(x)=\frac{1}{12}(\frac{\sqrt{x^{2}-4}}{x})^{3}[/tex]
 
  • #6
bLaf said:
Unfortunately, in some countries(like mine) the secant and cosecant trig functions are not taught(not even in mathematical schools).
Thanks!
I find it difficult to believe that these functions aren't presented in some countries. If that's the case, though, here is how they're defined:
secant(x) = sec(x) = 1/cos(x)
cosecant(x) = csc(x) = 1/sin(x)
 
  • #7
The Dagda said:
You could try [tex]u=2*sin^{-1}(x)[/tex]
Dagda, did you mean this as u = 2/sin(x)? That's different from 2*sin^(-1)(x), which is the same as 2*arcsin(x).
 
  • #8
Mark44 said:
Dagda, did you mean this as u = 2/sin(x)? That's different from 2*sin^(-1)(x), which is the same as 2*arcsin(x).

I think my brain exploded and I accidentally read the wrong trig sub off the table, whatever happened you are of course correct. :smile:
 

Related to Please give any hints about this integral

1. What is the best approach to solve this integral?

The best approach to solve an integral depends on the specific integral in question. Some common methods include substitution, integration by parts, and using trigonometric identities. It is important to choose a method that simplifies the integral and makes it easier to solve.

2. How do I determine the limits of integration for this integral?

The limits of integration for an integral are typically given in the problem or can be determined by the context of the problem. If the limits are not explicitly stated, they can be found by solving the equation that the integral represents.

3. Can you provide any tips for solving this integral?

One tip for solving an integral is to always check for any algebraic simplifications or trigonometric identities that can be used to make the integral easier to solve. It is also helpful to practice and become familiar with different integration techniques and when they are most useful.

4. Is there a specific order of steps that I should follow to solve this integral?

There is no specific order of steps that must be followed to solve an integral. However, it is helpful to first simplify the integrand, then choose an appropriate integration method, and finally evaluate the integral using the given limits of integration.

5. Are there any common mistakes to avoid when solving integrals?

One common mistake when solving integrals is forgetting to add the constant of integration when evaluating the integral. It is also important to be careful with algebraic manipulations and to double check the solution after completing the integration process.

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