 |
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 | 1. Discuss your best strategy for determining which part of the integrand should be u and which part should be dv.
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| | 2. Integration by parts comes from the product rule for derivatives. Which
integration technique comes from the chain rule? Briefly discuss why there is no commonly used
integration technique derived from the quotient rule.
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| In exercises 3-32, evaluate the integrals.
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| 3.  | 4.  |
| 5.  | 6. 
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| 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. 
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| In exercises 33-36, evaluate the integral using
integration by parts (as we recommended in the text, “Try something!”).
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| 33.  | 34.  |
| 35.  | 36. 
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| 37. How many times would
integration by parts need to be performed to evaluate  (where n is a positive integer)?
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| 38. How many times would
integration by parts need to be performed to evaluate  (where n is a positive integer)?
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| 39. Several useful
integration formulas called reduction formulas are used to automate the process of performing multiple integrations by parts. Prove that for any positive integer n,  (Use
integration by parts with u = cos n-1x and dv = cos x dx.)
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| 40. Use
integration by parts to prove that for any positive integer n, 
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| In exercises 41-48, evaluate the integral using the reduction formulas from exercises 39 and 40 and (2.4).
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| 41.  | 42.  |
| 43.  | 44.  |
| 45.  | 46.  |
| 47.  | 48. 
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| 49. Based on exercises 46-48, conjecture a formula for  (Note: You will need different formulas for m odd and for m even).
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| 50. Conjecture a formula for  .
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| 51. The excellent movie Stand and Deliver tells the story of mathematics teacher Jaime Escalante, who developed a remarkable AP calculus program in inner-city Los Angeles. In one scene, Escalante shows a student how to evaluate the integral . He forms a chart like the following:
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| |
sin x |
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| x2 |
- cos x |
+ |
| 2x |
- sin x |
- |
| 2 |
cos x |
+ |
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| Multiplying across each full row, the antiderivative is -x2cos x+2xsin x+2cos x+c. Explain where each column comes from and why the method works on this problem.
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| In exercises 52-57, use the method of exercise 51 to evaluate the integral.
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| 52.  | 53. 
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| 54.  | 55.  |
| 56.  | 57. 
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| 58. You should be aware that the method of exercise 51 doesn't always work, especially if both the
derivative and antiderivative columns have powers of x. Show that the method doesn't work on  .
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| 59. Show that  for positive integers m  n.
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| 60. Show that  for positive integers m n.
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| 61. Show that  for positive integers m and n.
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| 62. Show that  for any positive integer n.
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 | 63. Integration by parts can be used to compute coefficients for important functions called Fourier series. We cover Fourier series in detail in
Chapter 8. Here, you will discover what some of the fuss is about. Start by computing  for an unspecified positive integer n. Write out the specific values for a1, a2, a3 and a4 and then form the
function
f (x) = a1sin x+a2sin 2x+a3sin 3x+a4sin 4x.
Compare the graphs of y = x and y = f (x) on the interval [- , ]. From writing out a1 through a4, you should notice a nice pattern. Use it to form the
function
g(x) = f (x)+a5sin 5x+a6sin 6x+a7sin 7x+a8sin 8x.
Compare the graphs of y = x and y = g(x) on the interval [-, ]. Is it surprising that you can add sine functions together and get something close to a straight line? It turns out that Fourier series can be used to find cosine and sine approximations to any
continuous function on a closed interval.
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| | 64. Along with giving us a technique to compute
antiderivatives, integration by parts is very important theoretically. In this context, it can be thought of as a technique for moving derivatives off of one
function and onto another. To see what we mean, suppose that f (x) and g(x) are functions with f (0) = g(0) = 0, f (1) = g(1) = 0 and with continuous second derivatives f '' (x) and g'' (x). Use
integration by parts twice to show that 
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