Trigonometric Techniques of Integration
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7.3   
1. Suppose a friend in your calculus class tells you that this section just has too many rules to memorize. (By the way, the authors would agree.) Help your friend out by making it clear that each rule indicates when certain substitutions will work. In turn, a substitution u(x) works if the expression u' (x) appears in the integrand and the resulting integral is easier to integrate. In each of the rules covered in the text, identify u' (x) and point out why n has to be odd (or whatever the rule says) for the remaining integrand to be workable. Without memorizing rules, you can remember a small number of potential substitutions and see which one works for a given problem.
2. In the text, we suggested that when the integrand contains a term of the form, you might try the trigonometric substitution x = 2sin . We should admit now that this does not always work. How can you tell whether or not this substitution will work?
In exercises 3-40, evaluate the integrals.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
In exercises 41 and 42, evaluate the integral using both substitutions u = tanx and u = secx and compare the results.
41. 42.
43. Show that for any integer n > 1,
44. Evaluate (a), (b) and (c)
45. In an AC circuit, the current has the form i(t ) = Icos ( t ) for constants I and . The power is defined as Ri2 for a constant R. Find the average value of the power by integrating over the interval [0, 2 / ].
46. The area of the ellipse is given by . Compute this integral.
47. Evaluate the antiderivatives in Examples 3.2, 3.3, 3.5, 3.6 and 3.7 using your CAS. Based on these examples, speculate whether or not your CAS uses the same techniques that we do. In the cases where your CAS gives a different antiderivative than we do, comment on which antiderivative looks simpler.
48. Repeat exercise 47 for Examples 3.9, 3.10 and 3.11.
49. One CAS produces -sin2 x cos5x-cos5 x as an antiderivative in Example 3.2. Find c such that this equals our antiderivative of -cos5 x+cos7x+c.
50. One CAS produces -tan x-sec2xtan x+sec4xtan x as an antiderivative in Example 3.7. Find c such that this equals our antiderivative of tan3x+tan5x+c.
51. In Section 7.2, you were asked to show that forpositive integers m and n with m n, each of the following integrals equals 0. Also, Finally, for any positive integers m and n. We will use these formulas to explain how a radio can tune in an AM station. Amplitude modulation (or AM) radio sends a signal (e.g., music) that modulates the carrier frequency. For example, if the signal is 2sin t  and the carrier frequency is 16, then the radio sends out the modulated signal 2sin tsin 16t . The graphs of y = 2sin t , y = -2sin t  and y = 2sin tsin 16t  are shown in the figure.
The graph of y = 2sin tsin 16t  has the same period as the carrier sin 16t , but the amplitude varies between 2sin t  and -2sin t  (hence the term amplitude modulation). The radio's problem is to tune in the frequency 16 and recover the signal 2sin t . The difficulty is that other radio stations are broadcasting simultaneously. A radio receives all the signals mixed together. To see how this works, suppose a second station broadcasts the signal 3sin t  at frequency 32. The combined signal that the radio receives is 2sin tsin 16t+3sin tsin 32t . We will decompose this signal. The first step is to rewrite the signal using the identity

sin Asin B = cos (B-A)-cos (B+A).

The signal then equals

f (t ) = cos 15t-cos 17t+cos 31t-cos 33t.

If the radio “knows” that the signal has the form csin t  for some constant c, it can determine the constant c at frequency 16 by computing the integral and multiplying by 2/. Show that so that the correct constant is c = (2/) = 2. The signal is then 2sin t . To recover the signal sent out by the second station, compute and multiply by 2/. Show that you correctly recover the signal 3sin t .