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You are not logged in. #1 20060828 12:00:29
Integration by SubstitutionEvaluate the following integrals by the method of integration by substitution. Starred exercises may take more problem solving and manipulation than the others. Double starred problems should only be attempted by those who are quite experienced with the Calculus, and may cause anger and frustration. Triple starred problems are reserved for the truly insanely skilled. The beauty of the solution of triple starred problems combined with the sense of accomplishment is a true reward for the hard work put into the problem. Last edited by Zhylliolom (20060829 08:32:46) #2 20060828 12:50:07
Re: Integration by Substitution"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #3 20060828 13:27:02
Re: Integration by SubstitutionGood work Ricky, but as you must know, no credit is given for stating only the answer and not the solution method. #4 20060828 13:45:29
Re: Integration by SubstitutionUgh, you're going to make me type all that up? Fine... Since 3^{4x^2} is symmetric across the yaxis. Let: Then: Changing to polar coordinates: Standard integration follows, we get: Since 1/81 < 1, 1/81^a^2 approaches 0 as a approaches infinity. So: So the answer is: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20060828 14:02:39
Re: Integration by SubstitutionWait a minute, I never had to use integration by parts... "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #6 20060828 14:32:21
Re: Integration by Substitutionsubstitution* Last edited by Zhylliolom (20060830 07:15:45) #7 20090704 14:42:13
Re: Integration by Substitution1. Let u = 1+x^2 Last edited by glenn101 (20090704 14:44:00) "If your going through hell, keep going." #8 20090724 21:34:26
Re: Integration by SubstitutionHi! where the substitutions used have been z =  ln(x) first, and t^2 = z next. Jose Last edited by juriguen (20090724 21:35:08) “Make everything as simple as possible, but not simpler.”  Albert Einstein #9 20090724 21:50:05
Re: Integration by SubstitutionI found 16 is indeed much easier this way: where for the first step I have used x = exp(ln(x)), and the substitution is then 4x^2 ln(3) = t^2. Finally, the last step is done evaluating the erf function. Jose “Make everything as simple as possible, but not simpler.”  Albert Einstein #10 20090724 23:44:07
Re: Integration by SubstitutionNow 18: using 2  x = z^2 ! “Make everything as simple as possible, but not simpler.”  Albert Einstein #11 20090725 01:45:00
Re: Integration by SubstitutionFor #15 Start with the first one and say: Now for the second integral: Say: So Last edited by bobbym (20090725 02:06:21) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #12 20090725 02:01:11
Re: Integration by SubstitutionI would like to propose another integral, which took me really long to solve! (I would grade it at least with **) where Enjoy! “Make everything as simple as possible, but not simpler.”  Albert Einstein #13 20090725 03:51:57
Re: Integration by SubstitutionZhylliolom “Make everything as simple as possible, but not simpler.”  Albert Einstein #14 20090726 02:33:33
Re: Integration by SubstitutionThree easy ones; Last edited by bobbym (20090727 01:38:25) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #15 20090726 17:46:35
Re: Integration by SubstitutionThis is the only way I see for 19, but it is a little weird! where the substitution used has been alpha x^2 = u^2 Then, Jose Last edited by juriguen (20090726 21:08:29) “Make everything as simple as possible, but not simpler.”  Albert Einstein #16 20090730 10:14:36
Re: Integration by SubstitutionHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #17 20090813 21:09:02
Re: Integration by SubstitutionAnd another one: Last edited by Identity (20090813 21:11:56) #18 20090814 21:08:34
Re: Integration by SubstitutionHi Identity #19 20090815 00:57:44
Re: Integration by SubstitutionNice solution rzaidan, but I think the step of multiplying by (secx + tanx) requires a big leap of faith whereas usubstitution does not. #20 20090816 00:12:30
Re: Integration by SubstitutionHi Identity #21 20091124 18:03:37
Re: Integration by SubstitutionHi. This is my first time posting here. #22 20091124 18:10:55
Re: Integration by SubstitutionHi Denominator; Last edited by bobbym (20091124 18:12:33) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #23 20091125 04:53:34
Re: Integration by SubstitutionOkay my bad. #24 20091125 09:50:15
Re: Integration by SubstitutionOoops sorry rzaidan I was going to post my solution but I forgot Last edited by Identity (20091125 09:50:48) #25 20100929 05:13:18
Re: Integration by Substitution
Ans  0 