Appendix General rules of integration* (cont.) Ύ sech uduϭ sin Ϫ1 (tanh u)or2tan Ϫ1 e u Ύ du u 2 Ϫ a 2 ϭ 1 2a ln ΂ u Ϫ a u ϩ a ΃ ϭϪ 1 a coth Ϫ1 u a u 2 Ͼ a 2 Ύ csch uduϭ ln tanh u 2 or Ϫcoth Ϫ1 e u Ύ du a 2 Ϫ u 2 ϭ 1 2a ln ΂ a ϩ u a Ϫ u ΃ ϭ 1 a tanh Ϫ1 u a u 2 Ͻ a 2 Ύ sech 2 uduϭ tanh u Ύ du Ί a 2 Ϫ u 2 ϭ sin Ϫ1 u a Ύ csch 2 uduϭϪ coth u Ύ du Ί u 2 ϩ a 2 ϭ ln(u ϩ Ί u 2 ϩ a 2 ) or sinh Ϫ1 u a Ύ tanh 2 uduϭ u Ϫ tanh u Ύ du Ί u 2 Ϫ a 2 ϭ ln(u ϩ Ί u 2 Ϫ a 2 ) Ύ coth 2 uduϭ u Ϫ coth u Ύ du u Ί u 2 Ϫ a 2 ϭ 1 a sec Ϫ1 Ϳ u a Ϳ Ύ sinh 2 uduϭ sinh 2u 4 Ϫ u 2 ϭ 1 2 (sinh u cosh u Ϫ u) Ύ du u Ί u 2 ϩ a 2 ϭϪ 1 a ln ΂ a ϩ Ί u 2 ϩ a 2 u ΃ Ύ cosh 2 uduϭ sinh 2u 4 ϩ u 2 ϭ 1 2 (sinh u cosh u ϩ u) Ύ du u Ί a 2 Ϫ u 2 ϭϪ 1 a ln ΂ a ϩ Ί a 2 Ϫ u 2 u ΃ Ύ sech u tanh uduϭϪ sech u Ύ f (n) gdxϭ f (nϪ1) g Ϫ f (nϪ2) gЈ ϩ f (nϪ3) gЉ Ϫ иии (Ϫ1) n Ύ fg (n) dx Ύ csch u coth uduϭϪ csch u This is called generalized integration by parts. Ύ du u 2 ϩ a 2 ϭ 1 a tan Ϫ1 u a *Here, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated; e ϭ 2.71828 . . . is the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u Ͼ 0 [in general, to extend formulas to cases where u Ͻ 0 as well, replace ln u by ln ȊuȊ]; all angles are in radians; all constants of integration are omitted but implied. SOURCE : Murray R. Spiegel and John Liu, Mathematical Handbook of Formulas and Tables, 2d ed., Schaum’s Outline Series, McGraw-Hill, 1999. 638 Appendix Schematic electronic symblols* *From S. Gibilisco, The Illustrated Dictionary of Electronics, 8th ed., McGraw-Hill, 2001. 639 Appendix 640 Appendix 641 Appendix 642 Appendix 643 . symblols* *From S. Gibilisco, The Illustrated Dictionary of Electronics, 8th ed., McGraw-Hill, 2001. 639 Appendix 640 Appendix 641 Appendix 642 Appendix 6 43 . p, q, n any constants, restricted if indicated; e ϭ 2 .71 828 . . . is the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u Ͼ 0 [in general, to extend. constants of integration are omitted but implied. SOURCE : Murray R. Spiegel and John Liu, Mathematical Handbook of Formulas and Tables, 2d ed., Schaum’s Outline Series, McGraw-Hill, 1999. 638 Appendix Schematic