Langara College - Department of Mathematics and Statistics

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Internet Resources for the Calculus Student

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Techniques of Integration

Rather than using the Riemann Sum approximation, or other Numerical Methods,
we can often find definite integrals by using Antiderivatives.
And we can find many antiderivative rules by reading derivative rules "backwards"
(eg as when we use the fact that d/dx(x^2)=2x to see that x^2 is an antiderivative
for 2x). Combining these rules with some algebraic tricks gives us a substantial
arsenal of "Techniques of Integration".

See Techniques
of Integration : Substitution and Techniques
of Integration : Integration by Parts, from the extensive S.O.S.
MATHematics site.

This lesson on Trigonometric
Integrals comes from D.Hart at Indiana University.

An algebraic trick that is often useful for dealing with integration
of rational functions is to express them as sums of simpler terms by means
of the "Partial Fractions" decomposition.

See The
Method of Partial Fractions, from S.O.S.
MATHematics.

The University of Saskatchewan's Exercises
in Math Readiness (EMR) site also has a section on Partial
Fraction Decompositions. Included are an introduction and three sets
of exercises : Introductory,
Moderate,
and Advanced.

Despite all these tools, finding exact antiderivatives is not always
easy - or even possible!

Some people enjoy the "game" or "puzzle" aspect of this subject, and
even if you don't enjoy it, an understanding of the techniques can be important
for understanding how and why things work in various applications of the
calculus.

But if you just want to find or check an answer, then for many cases
it may be more convenient to use a computer algebra system such as Mathematica
(which will do integrals for you on-line at its "Integrator"
web site), or Maple (for which there is an on-line
interface at SFU).

Our "raw list" of links without
comment may include more sites of interest that we haven't yet referred
to here.

If you have come across any other good web-based illustrations of these
and related concepts,

please do let
us know and we will add them here.