You’ll probably see this integral someday:
.
It looks so simple, and you think, “gee, I probably was supposed to memorize that” or “oh I can do that, it looks so easy.” But then you don’t remember the antiderivative and get stuck. Cause, heck, what can you do with just anyway? It doesn’t decompose into anything nicer.
The trick is to use integration by parts. Let’s look at it backwards:
If we were really smart or really lucky, we might approach this problem by saying
.
(Is this really true? Use the product rule to show it.) Then we can integrate both sides:
.
Now using the Fundamental theorem of calculus, the integral on the left is just equal to . Solving for
, we get
.
Yay!

2 responses so far ↓
Integration by Parts « Calculus: Problems and Solutions // August 5, 2009 at 9:30 am |
[...] 5, 2009 · Leave a Comment When we looked at integrating the natural logarithm, we used a little trick: we took the product [...]
David Woodford // November 19, 2009 at 5:53 am |
Interesting, I was taught to do it by writing lnx = 1lnx but this way seems much simpler.
Great site in general,
Dave