Unit 9: Methods of Integration
The main thrust of this unit will be to give a list of substitutions
that would otherwise probably not occur to you. Specifically, we have
the trigonometric substitutions, where we replace an
expression involving a square root with an expression full of cosines
and tangents. It would seem that we leap from the frying pan into the
fire, but we are saved by the fact that there are lots of ``cute''
things to be done to trigonometric integrals. Pay special attention
to the half-angle
identities, which are often used to reduce the exponents of
sines and cosines down to something reasonable.
To complement our fun with trigonometry, we pull another trick out of
high school mathematics and resurrect completing the square. We
need this since most of our formulas are designed for expressions
as opposed to the more general
So, we complete the square to have the neccesary portions of the
integrands appear in the proper form.
This presentation is deliberately short on details; said details are
all in the texts.
After completing this unit you should be able to integrate many more
functions than before. In fact, you should be able to integrate
any of the common integratable functions.
- Read Simmons Chapter 10
- Sorry, there are not yet any World Web math entires on this topic.
- Clearly, you are going to need to work lots of problems if you
are going to have any hope of remembering the dozen or so tricks
presented in this unit. Again, no problems are inherently better than
others, so work a nice variety.
- Take the Practice
Test, Xdvi or PDF.
- Ask your instructor for a unit test.
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Last modified August 1, 1998