[Logo]   Langara College - Department of Mathematics and Statistics 

Internet Resources for the Calculus Student

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.

Review Contents of Math&Stats Dep't Website ....... Give Feedback ....... Return to Langara College Homepage