Integration by Substitution
which looks a lot simpler. If we were to integrate the function we would carry out the replacements in the following way:
It is nice to present a difficult problem in detail and then have students work alone or in small groups to solve a similar problem. For example, the following problem could be presented on the board with clear explanations for each step:
Solve the Integral:
Substitutions should be tested by having students perform integrals that are simplified with a substitution. The following are good questions to get them warmed up:
Integration by Parts
To teach this it is nice to do a problem in detail and then have students work individually on a similar problem trying to recall your steps as they go. To teach integration by parts put a problem on the board, like:
After this the formal equations can be introduced and they are much more likely to be understood than if they are discussed without a concrete example.
3. Find the following integral:
Integration by Partial Fractions
This lesson describes a general means by which complicated fractions of polynomials, or rational functions, can be written out as a sum of simpler fractions. Often one can use a guess and check technique to find the correct partial fraction decomposition. As a very simple example, consider the fraction:
However, adding these we obtain:
The guess and check technique is only of limited value for more complicated fractions and is only possible with some experience. Therefore, it will be important for students to practice with a number of examples. However, it should be clear that the integration part of these problems is really not important to the concept itself of partial fraction decomposition. So it is recommended that students are given the opportunity to gain experience with partial fractions without having to integrate them afterwards.
This can be done with a game. Have students take out only paper and pencil and divide the class into two teams. On the board you can write some rational function like:
and then call on the student who believes s/he has decomposed this correctly for his/her team. The student will have to present the work on the board, and if it is correct:
Again, it is not important that students complete the integration in these problems to learn and master the technique of partial fractions. Of course, this will eventually be important and should ideally be practiced. However if time is limited then the decomposition itself can be honed.
The material presented here is more or less a recipe for various possible combinations of sines, cosines, tangents and secants. Without some motivation, most students will find this lesson to be fairly dull. So it is recommended that a solid example of some kind be used to bring this to life. For example, one might discuss the voltage being in an electrical wall socket. This voltage is actually described by the function:
To understand this it is important to have a look at the plot for voltage above:
So the average voltage produced by any wall-socket is actually zero!
What we really want instead is a measure of how big the voltage is on average, and one way to do that is to make it positive everywhere and then take the average. This could be done with absolute values, but the simpler way is by squaring the function, taking its average, and then taking the square root of the result:
This function will not have an average of zero since it is everywhere positive:
and in fact we need to use integration of a power of sine here to obtain:
For example, if there is a part of the integral that involves a term like:
Similarly, if a part of the integral involves a term like:
It may be best to review quickly where the trig identities that are used in these substitutions come from. If students are encouraged to maintain a picture of a triangle in their minds, then it will not be difficult to recall each identity:
Then the teaching may be done by carrying out a few detailed calculations and requiring students to perform similar work just after. They should try to focus on your logic in each step instead of taking notes or memorizing formulas. That way when they attack the problems on their own the struggle to recall your logic will make the knowledge longer lasting.
Improper integrals can be described as simply a two-step process: We perform the integrals for values that we can do, and then look at the limit as we approach the values we’re not sure about. When a limits itself is infinite, we replace that limit by an arbitrary letter and then after we are done we let that letter go to infinity. When the integral passes over a point of infinite discontinuity then we simply replace that point by an arbitrary letter and then look at its limit after solving the integral again.
we find that it does converge to a finite number. Points of infinite discontinuity can be treated in a similar way.
As a general rule, giving students examples that have concrete applications will make this material more interesting. Students who may not be as motivated as others will appreciate a little less abstraction wherever possible.
Ordinary Differential Equations
Notice that the right hand side of the equation for a Linear ODE:
A discussion on the numerical methods for solving the differential equations follows naturally from a good example. Before actually applying some analytic trick it is a good idea to see where certain points take you along the numerical approximations. This can be very effectively done by tracking your progress along a plot of the isoclines like above. One numerical solution will generally not follow a single isocline, but as long as the step-size is small and the starting point is not near any major singularities, it should very closely follow the contour of an isocline.
Students can be tested on this material by being given simple differential equations to solve that closely mimic problems already solved in the text or in class. They should be encouraged to work in groups and to look for problems that look similar. In following the work and changing it as needed they will become experts at simple ODEs in no time at all.