Difficult integration: e^u(x^2)

In summary, The integral of the function g(x)=e^u(x) where u(x) = -(1-x^2)^(-1) from -1 to 1 is needed for constructing a Dirac function. There is no known way to solve this integral, including using the Cauchy Integral Formula and the chain rule. Mathematica gives an answer in terms of Meijer G functions, but this seems excessive.
  • #1
Kruger
214
0

Homework Statement



Given the function g(x)=e^u(x) where u(x) = -(1-x^2)^(-1). I have to integrate this from -1 to 1.

The Attempt at a Solution



I know the function is symmetric. It is enough to integrate it from 0 to 1 to get the real value of the integral. Well, beside that I have absolutely no clue how to do that. I need this in order to construct out of it a Dirac function. But my first task, as the homework states, is to solve this integral. (I tried to substitute something (but failed) and after that I wanted to use the Cauchy Integral Formule (extend the function to complex plane), but this didn't work either (because I couldn't get it in a appropriate form, as for the CIF needed)).

So I would be very pleased if someone can give me a hint. Perhaps a little bit more than a hint.
 
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  • #2
I don't think there is a way to do it. Neither does Mathematica. (Well, it gives an answer in terms of Meijer G functions, but that seems excessive.)
 
  • #3
maybe somehow incorporate the chain rule?
 

1. How do you integrate e^u(x^2)?

Integrating e^u(x^2) can be done using the substitution method. Let u = x^2, then du = 2x dx. This can be simplified to e^u du/2, which can then be integrated as e^u/2 + C. Finally, substitute back x^2 for u to get the final answer of e^(x^2)/2 + C.

2. What is the trick to solve difficult integrals involving e^u(x^2)?

The trick to solving these integrals is to recognize that e^u(x^2) is a composite function, where u is the exponent and x^2 is the base. Using the chain rule, we can rewrite the integral as e^u * (2x). This makes it easier to integrate using the substitution method.

3. Can you use integration by parts for e^u(x^2)?

Yes, integration by parts can also be used to solve integrals involving e^u(x^2). However, it may result in a more complex integral that still requires the substitution method to solve.

4. Are there any special cases when integrating e^u(x^2)?

One special case is when the limits of integration are from 0 to infinity. In this case, the integral becomes a definite integral and the final answer can be evaluated using the limit as x approaches infinity. Another special case is when u = -1, which results in the integral becoming 1/x^2, a well-known integral with a known solution.

5. What are some real-life applications of integrating e^u(x^2)?

Integrals involving e^u(x^2) are commonly used in statistical analysis and probability, especially in the field of physics, to determine the probability of certain events occurring. They are also used in finance and economics to calculate the expected value and risk of investments. In engineering, these integrals are used to model and analyze systems with complex variables.

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