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May 11th, 2017, 02:51 PM   #1
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Integral

Could you please solve that integral step by step

integral from 0 to 1 : ln(x^2+1)/(x+1)

Also, I attached as a photo..
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May 12th, 2017, 09:28 AM   #2
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I ran your problem through wolfram just to see if its easily solvable.

Wolfram|Alpha: Computational Knowledge Engine

It doesn't look simple, you may want to make sure you have it written correctly.
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May 12th, 2017, 07:07 PM   #3
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Yes, it is not simple.. And it has boundaries.. It has special solution, and that is what I want - if you are able to solve. - It should has something like that . Actually this question asked by our professor in college as a hard question, but he said that it has a trick, and solvable by some of the series.. You have to do a lot of times IBT(Integration By Part), and cut at somewhere, and find something with Series..
Answer is -π^2/48 + (3 log^2(2))/4 according to WolframAlpha: Wolfram|Alpha: Computational Knowledge Engine) .
After first IBT I got at the integral side : integral 2x[ln(x+1)/(x^2+1)]dx.
From 0 to 1 integral of ln(x+1)/(x^2+1) equals (ln(2)π)/8 but, because of I'm not an advance about IBT for definite integrals, I could not walk anymore..

Thanks in advance for your helps
Misty
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May 20th, 2017, 01:56 AM   #4
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Yes, it is hard!! There should be some tricks to solve it, and it has boundaries..

Thanks..
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May 25th, 2017, 10:18 AM   #5
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Any idea?
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May 25th, 2017, 12:44 PM   #6
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Special Integral

Do you still need this? I attempted a solution on paper, but stopped just before plugging in my bounds.

It may take me a bit to type the solution up.

Just let me know. Thanks.
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May 25th, 2017, 12:50 PM   #7
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I am interested in seeing a solution.
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May 27th, 2017, 09:46 AM   #8
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Try something like the following:

$\displaystyle \int_0^1\frac{\ln(x^2 + 1)}{x + 1}dx = \int_0^1\frac{\ln(x - i)}{x + 1}dx + \int_0^1\frac{\ln(x + i)}{x + 1}dx$

then finding those two integrals by writing the integrand as a series (possibly after making a substitution) and integrating that term by term.
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June 4th, 2017, 09:49 PM   #9
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Quote:
Originally Posted by thegrade View Post
Do you still need this? I attempted a solution on paper, but stopped just before plugging in my bounds.

It may take me a bit to type the solution up.

Just let me know. Thanks.
Yes I still need that..
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