Finding the Indefinite IntegralDate: 2/1/96 at 17:10:41 From: Anonymous Subject: integral calculus Dr. Math, I am a senior in an advanced high school calculus class. I am trying to find the indefinite integral of the following function: f(x)=x^x I have no clue as to what to do with this problem and I was wondering if you could help. -Matt Ford Date: 5/29/96 at 1:29:23 From: Doctor Pete Subject: Re: integral calculus There isn't a "nice" solution to the integral of f(x)=x^x. None of the substitutions or methods taught to evaluate integrals can completely integrate this function, and it is a general rule that most functions don't have integrals that are in "closed form," that is, the integral is some expression in other elementary functions. A well-known function with no closed form integral is e^(-x^2); however, the definite (improper) integral from 0 to infinity is closed. (Can you find its value?) On the other hand, one can compute the derivative f'(x) by logarithmic differentiation. We have f(x) = x^x, so ln(f(x)) = x*ln(x) and d(ln(f(x)))/dx = f'(x)/f(x) = 1+ln(x), so f'(x) = (1+ln(x))*x^x. -Doctor Pete, The Math Forum |
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