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Mathematics Tables MATHEMATICS TABLES Department of Applied Mathematics Naval Postgraduate School TABLE OF CONTENTS Derivatives and Differentials.............................................. 1 Integrals of Elementary Forms............................................ 2 Integrals Involving au + b ................................................ 3 Integrals Involving u2 a2 ............................................... 4 Integrals Involving √u±2 a2, a> 0 ....................................... 4 Integrals Involving √a2 ± u2, a> 0 ....................................... 5 Integrals Involving Trigonometric− Functions .............................. 6 Integrals Involving Exponential Functions ................................ 7 Miscellaneous Integrals ................................................... 7 Wallis’ Formulas ................................................... ...... 8 Gamma Function ................................................... ..... 8 Laplace Transforms ................................................... 9 Probability and Statistics ............................................... 11 Discrete Probability Functions .......................................... 12 Standard Normal CDF and Table ....................................... 12 Continuous Probability Functions ....................................... 13 Fourier Series ................................................... ........ 13 Separation of Variables .................................................. 15 Bessel Functions ................................................... ..... 16 Legendre Polynomials ................................................... 17 Trigonometric Functions in a Right Triangle ............................. 18 Trigonometric Functions of Special Angles ............................... 18 Complex Exponential Forms ............................................ 18 Relations among the Functions .......................................... 19 Sums and Multiples of Angles ........................................... 19 Miscellaneous Relations ................................................. 20 Inverse Trigonometric Functions ......................................... 20 Side-Angle Relations in Plane Triangles ................................. 21 Hyperbolic Functions ................................................... 22 First-Order Ordinary Differential Equations ............................. 23 Linear Second-Order Ordinary Differential Equations .................... 24 Vector Algebra ................................................... ....... 26 Vector Analysis ................................................... ...... 27 Fundamentals of Signals and Noise ...................................... 29 Version 2.7, July 2010 Initial typesetting: Carroll Wilde Graphics: David Canright Editing: Elle Zimmerman, David Canright Cover: in spherical coordinates (ρ, φ, θ), shell surface is ρ(φ, θ) = ekθ sin φ, 1 1+√5 where k = π ln G and G = 2 is the Golden Ratio. 1 FOREWORD This booklet provides convenient access to formulas and other data that are frequently used in mathematics courses. If a more comprehensive reference is needed, see, for example, the STANDARD MATHEMATICAL TABLES published by the Chemical Rubber Company, Cleveland, Ohio. DIFFERENTIALS AND DERIVATIVES A. Letters u and v denote independent variables or functions of an indepen- dent variable; letters a and n denote constants. B. To obtain a derivative, divide both members of the given formula for the differential by du or by dx. 1. d(a)=0 2. d(au)= a du 3. d(u + v)= du + dv 4. d(uv)= u dv + v du u v du u dv n n 1 5. d = − 6. d(u )= nu − du v v2 7. d(eu)= eudu 8. d(au)= au ln a du du du 9. d(ln u)= 10. d(log u)= u a u ln a 11. d(sin u) = cos u du 12. d(cos u)= sin u du − 13. d(tan u) = sec2 u du 14. d(cot u)= csc2 u du − 15. d(sec u) = sec u tan u du 16. d(csc u)= csc u cot u du − du du 17. d(arcsin u)= 18. d(arccos u)= √1 u2 −√1 u2 − − du 19. d(arctan u)= 20. d(sinh u) = cosh u du 1+ u2 du 21. d(cosh u) = sinh u du 22. d(arcsinh u)= √u2 +1 du 23. d(arccosh u)= , u> 1 √u2 1 − 24. (Differentiation of Integrals) If f is continuous, then u d( f(t) dt)= f(u) du. Za 25. (Chain Rule) If y = f(u) and u = g(x), then dy dy du = . dx du dx 2 INTEGRALS A. Letters u and v denote functions of an independent variable such as x; letters a, b, m, and n denote constants. B. Generally, each formula gives only one antiderivative. To find an expres- sion for all antiderivatives, add a constant of integration. ELEMENTARY FORMS 1. a du = au Z 2. af(u) du = a f(u) du Z Z 3. (f(u)+ g(u)) du = f(u) du + g(u) du Z Z Z 4. u dv = u v v du (Integration by parts) − Z Z un+1 5. un du = , n = 1 n +1 6 − Z du 6. = ln u u | | Z 7. eudu = eu Z au 8. audu = , a> 0 ln a Z 9. cos u du = sin u Z 10. sin u du = cos u − Z 11. sec2 u du = tan u Z 12. csc2 u du = cot u − Z 13. sec u tan u du = sec u Z 14. csc u cot u du = csc u − Z 3 15. tan u du = ln cos u − | | Z 16. cot u du = ln sin u | | Z 17. sec u du = ln sec u + tan u | | Z 18. csc u du = ln csc u cot u | − | Z du 1 u 19. = arctan u2 + a2 a a Z du u 20. = arcsin √a2 u2 a Z − du 1 u 21. = arcsec u√u2 a2 a a Z − INTEGRALS INVOLVING au + b 1 (au + b)n+1 22. (au + b)n du = , n = 1 a n +1 6 − Z du 1 23. = ln au + b au + b a | | Z u du u b 24. = ln au + b au + b a − a2 | | Z u du b 1 25. = + ln au + b (au + b)2 a2(au + b) a2 | | Z du 1 u 26. = ln u(au + b) b au + b Z 2(3 au 2 b) 3 27. u√au + b du = − (au + b) 2 15a2 Z u du 2(au 2b) 28. = − √au + b √au + b 3a2 Z 2 2 2 2 2(8b 12abu + 15a u ) 3 29. u √au + b du = − (au + b) 2 105a3 Z u2 du 2(8b2 4abu +3a2u2) 30. = − √au + b √au + b 15a3 Z 4 INTEGRALS INVOLVING u2 a2 ± du 1 u 31. = arctan [See 19] u2 + a2 a a Z du 1 u a 32. = ln − u2 a2 2a u + a Z − u du 1 33. = ln u 2 a2 u2 a2 2 | ± | Z ± u2 du a u a 34. = u + ln − u2 a2 2 u + a Z − u2 du u 35. = u a arctan u2 + a2 − a Z du 1 u2 36. = ln u(u2 a2) ±2a2 u2 a2 Z ± ± u du 1 37. = , n =0 (u2 a2)n+1 −2n(u 2 a2)n 6 Z ± ± INTEGRALS INVOLVING √u2 a2 , a> 0 ± u du 38. = u2 a2 [See 5] √u2 a2 ± Z ± p 1 2 2 3 39. u u2 a2 du = (u a ) 2 [See 5] ± 3 ± Z pdu 40. = ln u + u2 a2 √u2 a2 | ± | Z ± u p a2 41. u2 a2 du = u2 a2 ln u + u2 a2 ± 2 ± ± 2 | ± | Z p du 1 p u p 42. = ln u√u2 + a2 a a + √u2 + a2 Z du 1 u 43. = arcsec [See 21] u√u2 a2 a a Z − √u2 a2 u 44. − du = u2 a2 a arcsec u − − a Z √u2 + a2 p u 45. du = u2 + a2 + a ln u a + √u2 + a2 Z p 5 INTEGRALS INVOLVING √u2 a2 , a> 0 (Continued) ± u2 du u a2 46. = u2 a2 ln u + u2 a2 √u2 a2 2 ± ∓ 2 | ± | Z ± p 2 p 2 u 2 2 3 a u 47. u u2 a2 du = (u a ) 2 u2 a2 ± 4 ± ∓ 8 ± − Z p a4 p ln u + u2 a2 8 | ± | p INTEGRALS INVOLVING √a2 u2 , a> 0 − du u 48. = arcsin [See 20] √a2 u2 a Z − u a2 u 49. a2 u2 du = a2 u2 + arcsin − 2 − 2 a Z p u du p 50. = a2 u2 [See 5] √a2 u2 − − Z − p 1 2 2 3 51. u a2 u2 du = (a u ) 2 [See 5] − −3 − Z √pa2 u2 u 52. − du = a2 u2 + ln u − a + √a2 u2 Z p − du 1 u 53. = ln u√a2 u2 a a + √a2 u2 Z − − du √ a2 u2 54. = − u2√a2 u2 − a2u Z − √a2 u2 √a2 u2 u 55. − du = − arcsin u2 − u − a Z u2 du u a2 u 56. = a2 u2 + arcsin √a2 u2 − 2 − 2 a Z − p u a2u 57. u2 a2 u2 du = (a2 u2)3/2 + a2 u2 + − − 4 − 8 − Z p a4 u p arcsin 8 a 6 INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS u sin2au 58. sin2 audu = 2 − 4a Z u sin2au 59. cos2 audu = + 2 4a Z cos3 au cos au 60. sin3 audu = 3a − a Z sin au sin3 au 61. cos3 audu = a − 3a Z n 1 n sin − au cos au n 1 n 2 62. sin audu = + − sin − au du, − na n Z n 1 Z n cos − au sin au n 1 n 2 63. cos audu = + − cos − au du, na n Z Z u sin4au 64. sin2 au cos2 audu = 8 − 32a Z du 1 65. = cot au sin2 au −a Z du 1 66. = tan au cos2 au a Z 1 67. tan2 audu = tan au u a − Z 1 68. cot2 audu = cot au u −a − Z 1 1 69. sec3 audu = sec au tan au + ln sec au + tan au 2a 2a | | Z 1 1 70. csc3 audu = csc au cot au + ln csc au cot au −2a 2a | − | Z 1 u 71. u sin audu = sin au cos au a2 − a Z 1 u 72. u cos audu = cos au + sin au a2 a Z 2u a2u2 2 73. u2 sin audu = sin au − cos au a2 − a3 Z 2u a2u2 2 74. u2 cos audu = cos au + − sin au a2 a3 Z 7 INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS eau 75. ueau du = (au 1) a2 − Z eau 76. u2eau du = (a2u2 2au + 2) a3 − Z n au n au u e n n 1 au 77. u e du = u − e du, n> 1 a − a Z Z eau 78. eau sin budu = (a sin bu b cos bu) a2 + b2 − Z eau 79. eau cos budu = (a cos bu + b sin bu) a2 + b2 Z du 1 80. = u ln(1 + eau) 1+ eau − a Z MISCELLANEOUS INTEGRALS √1 a2u2 81. arcsin audu = u arcsin au + − a Z √1 a2u2 82. arccos audu = u arccos au − − a Z 1 83.
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