1.1. BASIC ANTI-DIFFERENTIATION 21 + C.

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1 .. BASIC ANTI-DIFFERENTIATION and so e x cos xdx = ex sin x + e x cos x + C. We end this section with a possibly surprising complication that exists for anti-di erentiation; a type of complication which does not occur for di erentiation. Remark..3. From the derivative formulas in [], we see that the derivative of any elementary functionis again an elementary function. You might hope that anti-derivatives/integrals would behave equally as well. They do not. It is easy to give elementary functions f(x) for which it is possible to prove that there is no elementary function F (x) such that F 0 (x) =f(x), i.e., f(x) has no elementary anti-derivative. Such functions f(x) includee x, e x,sin(x ), and cos(x ). This was first proved by Liouville in 835. The Fundamental Theorem of Calculus, Theorem.4.7, guarantees that the functions e x, e x,sin(x ), and cos(x ), and, in fact, all continuous functions, have some anti-derivative, but those anti-derivatives need not be elementary functions... Exercises Calculate the general anti-derivatives in Exercises through (4x +4x + 9) dx 5 w 5 sin t 7ew +6 3p w dw 3 p dt t v + p v v dv y + y dy + 5 3z 7 dz cos( ) d

2 CHAPTER. ANTI-DIFFERENTIATION: THE INDEFINITE INTEGRAL e p+4 dp r r 4 dr x p x + x p dx x (5! 3) 00 d! cos(t + 5) + 3 sin(9t) dt ln (x + ) x+5 dx e /x dx x 3 5 x ln x dx e /x 6x dx tan d 5 4+x dx t 4p t 5 +6dt x +4x +5 dx Hint: x +4x +5=(x + ) +. p x +6x 8 dx In each of Exercises through 3, find the anti-derivative of the given function which satisfies the given initial condition. The anti-derivative of each function with a lower-case name is denoted by the upper-case version of the same letter.. h(x) =4x +4x + 9, such that H( ) =. 3. p(w) = 5 w 7ew +6 3p w, such that P ( ) = q(t) =5sint 3 p, such that Q(0) = 7. t

3 .. BASIC ANTI-DIFFERENTIATION 3 5. k(v) = +v + p v v, such that K() =. 6. b(y) = y + y, such that B() = f(x) =x + x sin x, such that F ( ) =. 8. s(t) = t(ln t), such that S(e ) = g(x) =x p x +, such that G(0) =. 30. w(y) = tan y 4+y, such that W () =. 3. r(t) =te t t, such that R() = p. In each of Exercises 3 through 4, use integration by parts to find the indicated anti-derivative. 3. R xe 3x dx 33. R (x 5) e x dx 34. R t sin(t) dt 35. R t cos tdt 36. R p p ln pdp ln t 37. dt t 38. R e x sin xdx 39. R e x sin(5x) dx 40. R tan wdw 4. R w tan wdw Hint: At some point, you may want to use that w =(+w ). 4. Suppose that the net force F, acting on an object of mass m, pushes the object along the x-axis with an acceleration function, in m/s, of a(t) =sin(t), where t 0 is measured in seconds.

4 4 CHAPTER. ANTI-DIFFERENTIATION: THE INDEFINITE INTEGRAL a. Recall that F = ma and that momentum p = mv. Find the momentum of the object, as a function of time, if the mass of the object is 0 kilograms, and momentum at time t = 0 is 0 kilogram meters per second. b. What is the momentum of the object at time t = 4 seconds? 43. Repeat the preceding problem with the new acceleration function a(t) = t sin(t). 44. For each positive integer n, definef n ( ) =sin n cos and g n ( ) = cos n sin a. Find f n ( ) d and g n ( ) d. b. Find the specific anti-derivatives F n ( ) and G n ( ) that satisfy the initial conditions F n (0) = 5 and G n = 4. In Exercises 45 through 48, you are given the velocity of a particle at time t, and the position p(t 0 ) of the particle at a specific time t 0. Find the position function. 45. v(t) =3t 4t + 3, p() = v(t) =t + cos(t), p(0) = v(t) =t p 8 + 7t, p() = v(t) =at + b, p() = 5. Leave your answer in terms of a and b. 49. In the following steps, you will calculate the general anti-derivatives for sin x and cos x. a. Apply integration by parts to sin x sin xdx (written suggestively). b. Integration by parts yields a new anti-derivative. Use a trigonometric identity to write this new anti-derivative in terms of sin x, and solve your integration by parts equation for sin xdx. c. What is cos xdx? Hint: Use your answer to part (b). d. From the cosine double angle formula, cos(x) = cos x. Use this to integrate cos x, and explain why the di erent-looking answer that you obtain is, in fact, the same as your answer from part (c). Exercises 50 and 5 show that the argument in Exercise 49 can be generalized to calculate anti-derivatives of higher powers of sin and cos.

5 .. BASIC ANTI-DIFFERENTIATION a. Use integration by parts to prove that sin n tdt= n cos t sinn t + n n sin n tdt. Assume n. b. Use this formula to calculate previous problem. sin tdt. Check your answer by comparing to the 5. a. Use integration by parts to prove that cos n tdt= n cosn t sin t + n n cos n tdt. b. Use the formula in part (a) to determine cos tdt. 5. Suppose that instantaneous rate of change, with respect to time, of a population of an island at time t, measured in years, where t = 0 corresponds to the year 000, is given by p 6, 50, t 500 (t + ). The population of the island in 000 is a. Find an explicit formula for the population at time t. b. What is the predicted population in 050? 53. Prove that the argument used to calculate is di erentiable and prove that ln t dt can be generalized. Assume that f(t) f(t) dt = tf(t) tf 0 (t) dt. 54. Calculate e x sinh xdx. Hint: do not use integration by parts.

6 6 CHAPTER. ANTI-DIFFERENTIATION: THE INDEFINITE INTEGRAL 55. Consider the following logic in calculating e x sinh xdx using integration by parts. e x sinh xdx= e x sinh x = e x sinh x e x cosh xdx e x cosh x e x sinh xdx ) 0=e x sinh x e x cosh x ) e x sinh x = e x cosh x. Since e x > 0, we can divide and conclude that sinh x = cosh x. What is the flaw in this argument? 56. Prove the formula t n e t dt = t n e t n t n e t dt. Assume that n. 57. Calculate x dx, wherea 6= 0 is a constant. Hint: use Theorem 4..4 and an + a appropriate substitution. dx 58. Prove that p x + a =sinh x + C. a 59. Calculate p dx, wherea>0 is a constant. a x 60. Calculate x p x dx. Hint: consider sec x. 6. Calculate e x e ex dx. Hint: use substitution. 6. Let f (x) =e x and, for all integers n, let f n (x) =e fn (x). Prove that f (x) f (x)... f n (x) dx = f n (x)+c. 63. Calculate 64. Calculate 65. Calculate x dx. Hint: rewrite x as an exponential expression with base e. x +4 p x + dx. ln( + x ) dx.

7 .. BASIC ANTI-DIFFERENTIATION 7 e 3x 66. Calculate e x e x dx. Consider a simple electric circuit with an inductor, but no resistor or capacitor. A battery supplies voltage V (t). If inductance is constantly L, in henrys, then Kircho s Law from Example.7.8 of [] tells us that the current i at time t satisfies the di erential equation L di = V (t). In Exercises 67 through 70, find an dt explicit formula for i(t), given the condition i(t 0 )=i L =, V (t) =sint, i(0) = L = 9, V (t) =, i(3) =. 69. L = 3, V (t) = t t, i(0) = L = 3, V (t) = ln(/t) t, i() =. For each of the functions in Exercises 7 through 74, verify that (a) d apple f(x) dx = apple dx d f(x) and (b) dx f(x) dx = f(x)+c. Assume an appropriate domain for f(x). 7. f(x) =x f(x) = 3 cos(x). 73. f(x) =lnx. 74. f(x) = +x. 75. In the following steps, you will find the general anti-derivatives for functions of the form t n ln t. a. Suppose that n 6=. Apply integration by parts to find b. Find t ln tdt. t n ln tdt. 76. Suppose that water flows out of a hole 0. square meters in area from the bottom of a cylindrical tank with a base radius of meters and an initial height of 0 meters at a rate dv dt = t.3999 cubic meters per second.

8 8 CHAPTER. ANTI-DIFFERENTIATION: THE INDEFINITE INTEGRAL a. If the tank starts out full, what is the function V (t) for the volume in the tank at time t? b. Calculate the amount of water remaining in the tank one minute after the leak starts. c. Verify that dv dt = 0 precisely when the tank is empty. 77. A particle is traveling along the curve y = x, so that its x-coordinate is a function of time t, measured in minutes. Suppose that the horizontal velocity (i.e., velocity in only the horizontal direction) is given by dx/dt =0.5 cos 3 t miles per minute. a. What is the maximum horizontal speed (absolute value of velocity) on the time interval 0 apple t apple 0? b. Find the function x(t), the x-coordinate of the particle at time t, subject to the condition that the particle is at the origin at time t = 0. c. Find the function y(t), the y-coordinate of the particle at time t. d. What is the vertical velocity of the particle at time t = minutes? 78. An enclosed room is built in order to experiment with the e ects of pressure changes on objects. Suppose that the equipment is capable of decreasing the pressure in the room at a rate of t+ 0.04t kilopascals, kpa, per second, and that it can run for up to one minute before overheating. a. If the internal pressure in the room is 0.35 kpa (equal to the standard atmospheric pressure or atm), find an expression for P (t). b. How long will it take to reach 50 kpa (this is approximately the atmospheric pressure five and a half kilometers above the surface of the Earth)? c. What is the lowest pressure that can be reached in the room? In Exercises 79 through 8, you are given the acceleration, a = a(t) in m/s, of an object moving in a straight line, where t is the time in seconds. Find the velocity v and position p of the object, as functions of time, in terms of the initial velocity v 0 and initial position p a = t a =sint + cos t 8. a = e 3t 8. a = te t