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 | 1. It is never wrong to make a substitution in an integral, but sometimes it is not very helpful. For example, using the substitution u = x2 , you can correctly conclude that |
| but the new integral is no easier than the original integral. In this case, a better substitution makes this workable. (Can you find it?) However, the general problem remains of how you can tell whether or not to give up on a substitution. Give some guidelines for answering this question, using the integrals  and  as illustrative examples. |
| | 2. It is not uncommon for students learning substitution to use incorrect notation in the intermediate steps. Be aware of this - it can be harmful to your grade! Carefully examine the following string of equalities and find each mistake. Using u = x2 , |
| The final answer is correct, but because of several errors, this work would not earn full credit. Discuss each error and write this in a way that would earn full credit. |
| | 3. Suppose that an integrand has a term of the form ef (x). For example, suppose you are trying to evaluate  . Discuss why you should immediately try the substitution u = f (x). If this substitution does not work, what could you try next? (Hint: Think about  .) |
| | 4. Suppose that an integrand has a composite
function of the form f (g(x)). Explain why you should look to see if the integrand also has the term g' (x). Discuss possible substitutions. |
| In exercises 5-8, use the given substitution to evaluate the indicated integral.
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| 5. 
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| 6. 
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| 7. 
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| 8. 
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| In exercises 9-46, evaluate the indicated integral.
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| 9.  | 10.  |
| 11.  | 12.  |
| 13.  | 14. 
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| 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.  |
| 41.  | 42.  |
| 43.  | 44.  |
| 45.  | 46. 
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| In exercises 47-56, evaluate the definite integral.
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| 47.  | 48.  |
| 49.  | 50. 
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| 51.  | 52.  |
| 53.  | 54.  |
| 55.  | 56. 
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| In exercises 57-66, evaluate the integral exactly, if possible. Otherwise, estimate it numerically.
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| 57.  | 58.  |
| 59.  | 60.  |
| 61.  | 62.  |
| 63.  | 64.  |
| 65.  | 66. 
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| In exercises 67-70, make the indicated substitution for arbitrary functions
f (x).
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| |
| 67. u = x2 
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| 68. u = x3 for 
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| 69. u = sin x for 
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| 70. u =  for 
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| 71. A
function f (x) is said to be even if f (-x) = f (x) for all x. f (x) = x2 and f (x) = x4 are even, since (-x)2 = x2 and (-x)4 = x4.. A
function f (x) is said to be odd if f (-x) = -f (x). Show that cos x and xsin x are even, but sin x and xcos x are odd. |
| 72. Suppose f (x) is continuous for all x. For any positive
constant a,  . Using the substitution u = -x in  , show that if f (x) is even, then  . Also, if f (x) is odd, show that  . |
| In exercises 73-78, determine if the integrand is even or odd (or neither), rewrite the integral accordingly and compute (or estimate) the integral.
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| 73.  | 74.  |
| 75.  | 76.  |
| 77.  | 78. 
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| |
| 79. The location  of the center of gravity (balance point) of a flat plate bounded by y = f (x) > 0 , a  x b and the x - axis is given by  and  . For the semicircle  , use symmetry as in exercises 73-78 to argue that  and  . Compute  . |
| 80. Suppose that the population density of a group of animals can be described by  thousands of animals per mile for 0 x 2 , where x is the
distance from a pond. Graph y = f (x) and briefly describe where these animals are likely to be found. Find the total population  . |
| 81. The voltage in an AC (alternating current) circuit is given by V(t ) = Vpsin (2 ft) , where f is the
frequency. A voltmeter does not indicate the amplitude Vp. Instead, the voltmeter reads the root-mean-square (rms), the
square root of the average value of the square of the voltage over one cycle. That is, rms =  . Use the trigonometric identity sin 2x =  -cos 2x to show that rms = Vp/ . |
| 82. Graph y = f (t ) and find the root-mean-square of f (t ) =  where 
|
 | 83. A predator-prey system is a set of differential equations modeling the change in population of interacting species of organisms. A simple model of this type is |
| for positive constants a, b, c and d. Each equation includes a term of the form x(t )y(t ), which is intended to represent the result of confrontations between the species. Noting that the contribution of this term is negative to x'(t ) but positive to y' (t ) , explain why it must be that x(t ) represents the population of the prey and y(t ) the population of the predator. If x(t ) = y(t ) = 0 , compute x' (t ) and y'(t ). In this case, will x and y increase or decrease or stay constant? Explain why this makes sense physically. Determine x' (t ) and y' (t ) and the subsequent change in x and y at the so-called equilibrium point x = c/d , y = a/b. If the population is
periodic, we can show that the equilibrium point gives the average population (even if the population does not remain constant). To do so, note that  a-by(t ). Integrating both sides of the equation from t = 0 to t = T [the period of x(t ) and y(t ) ], we get  . Evaluate each integral to show that  Assuming that x(t ) has period T, we have x(T) = x(0) and so, 0 =  . Finally, rearrange terms to show that  ; that is, the average value of the population y(t ) is the equilibrium value y = a/b. Similarly, show that the average value of the population x(t ) is the equilibrium value x = c/d. |
