


In this part of the module we will begin by assuming that the functions under investigation have period 2 pi. As a first step in the process of breaking down complex periodic functions into simple sine and cosine functions we investigate definite integrals of the forms given below:
1. Evaluate the following integrals by hand. (Be careful in the cases where m = 0.)
(a)
(b)
(c)
(d)
2. We now investigate integrals of the form
(a)
(b)
(c)
(d)
3. What can you conclude about
4. Verify the conclusions you reached in answering Question 3 by evaluating appropriate integrals on your worksheet for a range of values of m and n.
5. Explain your answer to Question 3 in light of what you know about odd and even functions.
6. We now turn to integrals of the form
(a) What general conjecture can you make about
(b) Study the graph of y = sin(5t) sin(3t). Does this change the conjecture you made in (a)?
7. (a) Verify the conjecture you made in the last question by evaluating the integrals on your worksheet
with various values for m and n where m is not equal to n.
(b) We now look at a general formula for integrals of the form
(c) Plot the graphs of at least three functions of the form f(t) = sin(mt) sin(nt), where m = n.
Make a conjecture about the integrals in this case.
(d) Verify your conjecture made in (c) by using your computer algebra system to evaluate the
integrals of the functions you graphed and by analyzing the general formula.
8. Make a final conjecture about integrals of the form
9. Investigate integrals of the form



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