Integration by PartsDate: 12/8/95 at 5:12:43 From: "Juergen Meier" Subject: Integration can be difficult Suche Stammfunktion fuer : x^n/e^x Date: 5/30/96 at 11:16:51 From: Doctor Anthony Subject: Re: Integration can be difficult Using integration by parts, it is easy to get a reduction formula, especially if the range of integration is from 0 to infinity. I shall assume these limits for the integration Let I(n) = INT{x^n/e^x.dx} = INT{x^n.e^(-x).dx} (from 0 to infinity) = x^n(-e^(-x)) + n.INT{x^(n-1).e^(-x).dx} (from 0 to infinity) = 0 + n.I(n-1) and continuing in this manner = n(n-1).I(n-2) Continue this process until, n(n-1)(n-2)...3.2.1.INT{e^(-x).dx} (0 to inf.) and this final integral is +1 So the value of the integral is n(n-1)(n-2)...3.2.1 = n! If the limits are other than 0 to infinity, then there will be no simple formula for the integral. Provided n is not too large, it can nevertheless be evaluated quite quickly. -Doctor Anthony, The Math Forum |
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