Expert: Scotto - 7/20/2008

Question
QUESTION: I'm a Math teacher myself, and I'm mystified by what I thought was a simple problem: to integrate lnx between a pair of sensible limits (1 - 6).  (answer is around 5.7)
It falls out via integration by parts in a few lines, but the teacher who set the problem as part of a much larger assignment, has instructed his students that they are not to use either integration by parts or integration by substitution (which isn't very helpful here anyway).  Instead they are to solve it 'by recognition'!
My guess is that this teacher, who refuses to discuss the matter with his students or colleagues, has goofed and unconsciously confused integration and differentiation here and so thinks the answer is 1/x.  Presumably he hasn't bothered to check through his assignment before issuing it either.  His intransigence is of course wasting huge amounts of his students' time.
Am I being dense? do you know another simple solution to this integration?

ANSWER: It is known that the derivative of ln(x) is 1/x, not the integral.

The integral of ln(x) is found by the u-v method and is xln(x) - x evaluated from 1 to 6.

Evaluating this gives 1*ln(1)-1=-1 and 6*ln(6)-6 = 4.75.  

4.75- -1 = 5.75.

---------- FOLLOW-UP ----------

QUESTION: okay, but I'd guess by 'the u-v method' you mean the process of integration by parts, which this teacher has specifically prohibited.  Frankly it seems pretty silly to disallow a method that works, but the reason would be that this particular syllabus does not run to int-by-parts.  My problem is to do this 'by recognition', whatever he means by that.  I don't see any reasonable way that a fifteen-year-old student could be expected to more or less intuit this result.  I'm trying to find a way to defuse a potentially explosive professional situation, but can't find an approach other than the one you and I both know.

Answer
That's the only way I know how to do the problem.  If the answer is known it can be seen to be right. Maybe since the power of x is 0, you could take ln(x) and multiply by x.

You could then look at x*ln(x) as the answer.  The derivative of x*ln(x) is a product rule.  The answer is x/x+ln(x)=1+ln(x).  Since there is an extra 1 in the answer, there must be a minus x in the result to counteract this.

Seeing this, we now know that answer is x*ln(x) - x without doing any u=ln(x) dv=dx.  Maybe that's what he means by intuition.