Is integration impossible here?

Dhamnekar Winod

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Hi,
How to integrate \(\displaystyle\int_0^\infty \tanh{(\pi*x)}*\left(\frac1x-\frac{x}{\frac{1}{16}+x^2}\right)dx\) and then how to express it in this format \(\frac{\pi^a}{b}-\ln{(c)}\), where a, b, c are positive integers.


My attempt: We can rewrite this integral as \(\displaystyle\int_0^\infty \frac{\tanh{(\pi*x)}}{16*x^3+x}dx\). Now, how to proceed further?🤔

My offline integral calculator gives me this answer 0.877263492537. I don't understand how is that computed?:confused:

If any member knows the correct answer to this question, may reply to this question with correct answer.:)
 
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Dhamnekar Winod

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Hi,
More accurate answer for this question is 0.877649146234951 given by my offline integral calculator.
 

Dr.Peterson

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Where did the problem come from, and what is the exact wording? What did you try, that convinced you it can't be done without technology?
 

HallsofIvy

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First, in the strictest sense, yes, that is "integrable"- there exist some number equal to its integral. It does not necessarily have an "anti-derivative" in terms of elementary functions. As to "how that is computed", there are a number of numerical ways to calculate an integral, Simpson's rule, the midpoint rule, the trapezoid method, …. and I can't be sure which your "offline integral calculator" uses.

(I am puzzled as to why your "offline integral calculator" gave you two different results for the same problem.)
 

Singleton

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Generally, you are better off leaving the rational functions separated (as given), which allows you to break it up into a sum of two integrals. (There is even an integration strategy of "partial fractions" to separate rational functions.) So here you don't want to do your common denominator thing.

What is the advantage of the separated rational functions? They are of lower power. For one thing, it makes it easier to integrate by parts. With only terms of x or x^2 appearing, we begin to spot the derivatives and antiderivatives of some common functions such as ln x or trig functions.
 

Dr.Peterson

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Generally, you are better off leaving the rational functions separated (as given), which allows you to break it up into a sum of two integrals. (There is even an integration strategy of "partial fractions" to separate rational functions.) So here you don't want to do your common denominator thing.

What is the advantage of the separated rational functions? They are of lower power. For one thing, it makes it easier to integrate by parts. With only terms of x or x^2 appearing, we begin to spot the derivatives and antiderivatives of some common functions such as ln x or trig functions.
I would have suggested the same ideas, since the problem evidently expects an exact answer.

But Wolfram Alpha indicates that the pieces can't be integrated in terms of basic functions, and gives only the numerical definite integral for the whole thing.

This is why I want to see the original problem and its source.
 

Singleton

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I would have suggested the same ideas, since the problem evidently expects an exact answer.

But Wolfram Alpha indicates that the pieces can't be integrated in terms of basic functions, and gives only the numerical definite integral for the whole thing.

This is why I want to see the original problem and its source.
Oh ok sorry I was not careful. The OP lost the numerator over the common denominator. Actually the integrand is O(1/x) for x large isn't it? Bad decay. I am not familiar with Wolfram Alpha. Was it saying the integral does not converge at infinity or that the indefinite integral cannot be represented in closed form? Something is strange with the problem. In the ln(c) is c going to infinity?
 

Dr.Peterson

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Dhamnekar Winod

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Here's what it says for the first term: https://www.wolframalpha.com/input/?i=integral+of+tanh(pi+x)/x (says it can't be integrated in terms of standard functions, so your method wouldn't work)

Here's the whole thing: https://www.wolframalpha.com/input/?i=integral+of+tanh(pi+x)/(x(16x^2+1)) (only states the definite integral in numerical form, confirming OP's answer but suggesting it can't be done exactly)

I haven't tried very hard to do it, since I'm waiting for an answer from the OP.
Hi,

To solve this question, all supported integration methods have been tried unsuccessfully. Note that many functions don't have an elementary antiderivative. But i have the answer using 'Contour Integration technique' but i didn't understand it, as I have not studied contour integration in much details. So for the time being, the status of this question will remain as "Unsolved" from my point of view.
I don't know how to solve it using numerical integration methods. :)
 

Dr.Peterson

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To solve this question, all supported integration methods have been tried unsuccessfully. Note that many functions don't have an elementary antiderivative. But i have the answer using 'Contour Integration technique' but i didn't understand it, as I have not studied contour integration in much details. So for the time being, the status of this question will remain as "Unsolved" from my point of view.
I don't know how to solve it using numerical integration methods. :)
This is one reason we ask for the context of a question. Contour integration is a way to get an exact definite integral in a case like this, which I hadn't even considered because nothing you've said suggests it. Also, if you have an answer, why not show it to us so we can help explain it??

Are you saying this problem was assigned to you, but you have not learned anything about contour integration, or do you just not understand the details? Or did this come from somewhere other than a course you are taking?
 
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