Is There an Analytical Solution for This Tough Definite Integral?

In summary, the given definite integral evaluates to approximately 1.4695 numerically and appears to converge within the domain of 0 to 1. However, there does not seem to be an analytical solution for this integral and Wolframalpha confirms that it does not converge. The integrand becomes infinitely large near zero, making it larger than a divergent integral. With the correction of a minus sign, the integral should now converge.
  • #1
Irid
207
1
Hello,
I would like to evaluate the following definite integral

[itex] \int_0^1 \frac{exp(1/(x(1-x)))}{\sqrt{x(1-x)}} [/itex]

Numerically I get the result of about 1.4695 and it appears to converge nicely in the domain of interest (0;1). However, I'm wondering whether some kind of analytical integral exists as well... couldn't find anything myself :(
 
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  • #3
  • #4
Oops, sorry, I forgot the minus sign:

[itex] \int_0^1 \frac{exp(-1/(x(1-x)))}{\sqrt{x(1-x)}} [/itex]

should converge now :)
 

1. What is a "Tough definite integral"?

A tough definite integral is a type of mathematical problem that involves finding the area under a curve or the volume of a solid using integration. Unlike simpler integrals, tough definite integrals require more advanced techniques and may involve complicated functions.

2. How do I solve a tough definite integral?

To solve a tough definite integral, you need to use integration techniques such as substitution, integration by parts, or trigonometric identities. You may also need to break the integral into smaller parts or use numerical methods if the function is too complex.

3. What is the purpose of finding a tough definite integral?

The purpose of finding a tough definite integral is to calculate the area under a curve or the volume of a solid in real-world applications. This can help in solving problems related to physics, engineering, economics, and other fields that involve continuous functions.

4. Can a tough definite integral have multiple solutions?

No, a tough definite integral can only have one unique solution. This is because the integral represents the area or volume of a specific region, and there can only be one correct answer for a given function.

5. Is there a way to check if my solution to a tough definite integral is correct?

Yes, you can use the Fundamental Theorem of Calculus to check if your solution is correct. This theorem states that the derivative of the definite integral of a function is equal to the original function. You can also use technology, such as a graphing calculator, to visualize and verify your solution.

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