Find it's Closed Form

Calculus Level 5

$\int_{0}^{\infty} \dfrac{ \arctan x \cdot \ln(1+x^{2}) \cdot \sqrt{x}}{1+x^{2}} dx = \ ?$

Note: This is not an original question.

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The answer is 11.7433.

2 solutions

Lu Chee Ket
Feb 7, 2015

d x/ (1 + x^2) can only serve either d (arctan x) xor d (Ln (1 + x^2)), while not 2 x, Sqrt (x) seem to block them from being able to be integrated. Therefore, I applied numerical method to obtain 11.74 rather than 11.743 but quite all right.

Hard to satisfy both finest and also sufficient coverage for infinity. To compromise them, I found that coverage for infinity dominated. Due to constraint of limited speed allowed by computer, I actually set finest to only 0.1 and infinity with 1,000,000,000 for my whole verification, finally. As a consequence, I actually made 10,000,000,000 of n terms for numerical integration of summing numerical terms.

Integral = (0.1)/ 3 Sum (F + L + 4 E + 2 R). At very end I tried to estimate for result of 10 times coverage by (0.1)/ 3 [6 E] = 0.2 E as F = 0, L --> 0 while E and R are approximately equal to each other, to look for a stoppage of growth to indicate for a convergence. While I am still calculating where 11.73877 has been maximum obtained so far. Thinking that if 11.74 isn't the answer, then 11.8 ought to come by the end. Having 3 trials available, I tried to see if 11.74 works.

Fortunately, my 11.74 works! I don't need to worry for further work although I haven't really completed the task. This integral is quite special that its upper limit of infinity converged at quite a huge number. A column of Excel with 1048576 cells can sum up to only less than 11.70 which is tough and insufficient; hard to confirm by further check as mentioned. I got 11.74 by using Turbo Pascal.

Hi Lu Chee Ket,I think you should mention the topic which you are actually dealing with...i wasted my half an hour to solve this problem.....so in future please mention your topic...

Himanshu Parihar - 6 years, 2 months ago

Ronak Agarwal
Feb 15, 2015

@Azhaghu Roopesh M please at least comment whether closed form of this integral exists or not, I don't want wasting my time unneccasarily on find it's closed form.

I did this integral by numerical methods.

If you assure that closed form exists than it is an very interesting problem.

Actually , this question can be brought to an evaluable form . Let me make this clear, I used a complex numbers approach and have partially solved this question,but not completely .

This question was given to me by a friend from Resonance who gave me a hint of using complex numbers and Dirichlet Beta function . I'll be thankful if any of you could post a solution to this problem .

A Former Brilliant Member - 6 years, 3 months ago

So you gurantee a nice closed form in term of gamma, beta, zeta or whatever other known functions, then this would be nice question to work on.

@Azhaghu Roopesh M

Ronak Agarwal - 6 years, 3 months ago

Which Numerical Integration method did you apply ? Was it the rectangle rule.

Rajdeep Dhingra - 6 years, 3 months ago

Find it's Closed Form

Note: This is not an original question.

You can try more of my Questions here .

The answer is 11.7433.

2 solutions

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