My Math Forum Integral - compare solution

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 June 28th, 2012, 11:25 PM #1 Newbie   Joined: Jun 2012 Posts: 13 Thanks: 0 Integral - compare solution I have one integral and I would like comparing solution with you $\int \frac{1}{\left(x^2+4\right)\sqrt{1-x^2}} \, dx$
 June 29th, 2012, 12:13 AM #2 Newbie   Joined: Jun 2012 Posts: 13 Thanks: 0 Re: Integral - compare solution This is my solution $\frac{\sqrt{5}}{5}\text{ArcTan}\left(\frac{\sqrt{5 }}{2}\text{Tan}(\text{ArcSin}(X))\right)$ and this is Mathematica $-\frac{\text{ArcTan}\left[\frac{x \sqrt{5-5 x^2}}{2 \left(-1+x^2\right)}\right]}{2 \sqrt{5}}$
 June 29th, 2012, 01:43 AM #3 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Integral - compare solution Your answer needs to be halved. If you have doubts about any result, check with differentiation and then try graphing it. There are many ways in which an antiderivative can turn out, especially when you remember the constant may differ. For example, I got $\frac{\operatorname{sgn}\,x}{2\sqrt5}\,\operatorna me{arcsin}\,\sqrt{\frac{5x^2}{x^2+4}}$, which is just the same function if x is real.
 June 29th, 2012, 02:17 AM #4 Newbie   Joined: Jun 2012 Posts: 13 Thanks: 0 Re: Integral - compare solution I know that may be multisolution depend of way of integration. Graphical method is good but in case where we don't have calculator or program may be serious problem. In any cases I thank to you for effort in discovering solution.
June 29th, 2012, 05:08 PM   #5
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Re: Integral - compare solution

aswoods is correct . . . You need a 2 in the denominator.

Quote:
 $\text{This is my solution: }\:\frac{\sqrt{5}}{5}\text{ArcTan}\left(\frac{\sqr t{5}}{2}\text{Tan}(\text{ArcSin}(X))\right)$ $\text{and this is Mathematica: }\:-\frac{\text{ArcTan}\left[\frac{x \sqrt{5-5 x^2}}{2 \left(-1+x^2\right)}\right]}{2 \sqrt{5}}$

$\text{Let }\theta \,=\,\arcsin(x) \;\;\;\sin\theta \,=\,\frac{x}{1}$

$\theta\text{ is in a right triangle with: }\,opp = x,\;hyp = 1
\;\;\text{Hence: }\,adj = \sqrt{1\,-\,x^2} \;\;\;\Rightarrow\;\;\;\tan\theta \:=\:\frac{x}{\sqrt{1\,-\,x^2}}$

$\text{Your answer would have been: }\:\frac{1}{2\sqrt{5}}\,\arctan\left(\frac{\sqrt{5 }\,x}{2\sqrt{1\,-\,x^2}}\right)$

$\text{Mathematica's answer is rather stupid . . . }$

$\text{W\!e have: }\:-\frac{1}{2\sqrt{5}}\,\arctan\left(\frac{x\sqrt{5\,-\,5x^2}}{2(-1\,+\,x^2)}\right) \;=\;-\frac{1}{2\sqrt{5}}\,\arctan\left(\frac{x\sqrt{5(1 \,-\,x^2)}}{-2(1\,-\,x^2)}\right)$

[color=beige]. . [/color]$=\;\frac{1}{2\sqrt{5}}\,\arctan\left(\frac{x\,\cdo t\,\sqrt{5}\,\cdot\,\sqrt{1\,-\,x^2}}{2(1\,-\,x^2)}\right) \;=\;\frac{1}{2\sqrt{5}}\,\arctan\left(\frac{\sqrt {5}\,x}{2\sqrt{1\,-\,x^2}}\right)$

 June 30th, 2012, 03:15 AM #6 Senior Member   Joined: May 2011 Posts: 501 Thanks: 6 Re: Integral - compare solution Often times, math engines such as Maple and Mathematica will give unsimplified solutions. One can simplify them down by using 'simplify'. I have Maple, and often I have to use 'simplify(%)' to get it down to a reasonable looking solution. I am not sure what the code is in Mathematica.
 June 30th, 2012, 09:33 AM #7 Senior Member   Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0 Re: Integral - compare solution I noticed that the quality of Wolfram Alpha isn't very good. Although it can solve some problems nothing else can (when it chooses to), it's terrible at simplifying those answers. It used to have problems simplifying log(e).
June 30th, 2012, 10:50 AM   #8
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Re: Integral - compare solution

Quote:
 Originally Posted by The_Fool I noticed that the quality of Wolfram Alpha isn't very good. Although it can solve some problems nothing else can (when it chooses to), it's terrible at simplifying those answers. It used to have problems simplifying log(e).
[color=#000000]If you speak for the internet site then ok, it is only a small portion of what the commercial program can do. But if you speak for Wolfram-Mathematica you are 100% wrong.[/color]

 June 30th, 2012, 02:45 PM #9 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Integral - compare solution You have to remember that it tries to produce an answer that is valid for as many arguments as possible. For complex arguments, many obvious simplifications ignore whatever branch cuts are in use and so may result in a strictly incorrect answer. In many cases it should show underneath a simplified version "for real x" or "for positive x".
July 1st, 2012, 02:18 AM   #10
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Re: Integral - compare solution

Quote:
 Originally Posted by aswoods You have to remember that it tries to produce an answer that is valid for as many arguments as possible. For complex arguments, many obvious simplifications ignore whatever branch cuts are in use and so may result in a strictly incorrect answer. In many cases it should show underneath a simplified version "for real x" or "for positive x".
[color=#000000]There is a command that inputs such restrictions.
For example

Code:
Assuming[x>0,Integrate[Sqrt[x],x]]
.[/color]

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