Wednesday, 8 January 2014

calculus - Difficult Indefinite Integral



(logx+1)xxdx



This integral was found from the MIT Integration Bee. After making several unsuccessful attempts, I decided to type it into Mathematica, only to find that Mathematica could only produce an answer for this integral in the case where log(x) referred to the natural logarithmic function ln(x).


Answer



The form of the integrand suggests writing (1+logx)xx=(1+logx)exlogx,

then observing that by the product rule, ddx[xlogx]=x1x+1logx=1+logx.
Consequently, the integrand is of the form f(x)ef(x), and its antiderivative is simply ef(x)=exlogx=xx.



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