Review of di®erentiation and integration rules from Calculus I and II for Ordinary Di®erential Equations, 3301 General Notation: a; b; m; n; C are non-speci¯c constants, independent of variables e; ¼ are special constants e = 2:71828 ¢ ¢ ¢, ¼ = 3:14159 ¢ ¢ ¢ f; g; u; v; F are functions f n(x) usually means [f(x)]n, but f ¡1(x) usually means inverse function of f a(x + y) means a times x + y, but f(x + y) means f evaluated at x + y fg means function f times function g, but f(g) means output of g is input of f t; x; y are variables, typically t is used for time and x for position, y is position or output 0;00 are Newton notations for ¯rst and second derivatives. d d2 d d2 Leibnitz notations for ¯rst and second derivatives are and or and dx dx2 dt dt2 Di®erential of x is shown by dx or ¢x or h f(x + ¢x) ¡ f(x) f 0(x) = lim , derivative of f shows the slope of the tangent line, rise ¢x!0 ¢x over run, for the function y = f(x) at x General di®erentiation rules: dt dx 1a- Derivative of a variable with respect to itself is 1. = 1 or = 1. dt dx 1b- Derivative of a constant is zero. 2- Linearity rule (af + bg)0 = af 0 + bg0 3- Product rule (fg)0 = f 0g + fg0 µ ¶ f 0 f 0g ¡ fg0 4- Quotient rule = g g2 5- Power rule (f n)0 = nf n¡1f 0 6- Chain rule (f(g(u)))0 = f 0(g(u))g0(u)u0 7- Logarithmic rule (f)0 = [eln f ]0 8- PPQ rule (f ngm)0 = f n¡1gm¡1(nf 0g + mfg0), combines power, product and quotient 9- PC rule (f n(g))0 = nf n¡1(g)f 0(g)g0, combines power and chain rules 10- Golden rule: Last algebra action speci¯es the ¯rst di®erentiation rule to be used Function-speci¯c di®erentiation rules: (un)0 = nun¡1u0 (uv)0 = uvv0 ln u + vuv¡1u0 (eu)0 = euu0 (au)0 = auu0 ln a u0 u0 (ln(u))0 = (log (u))0 = u a u log a (sin(u))0 = cos(u)u0 (cos(u))0 = ¡ sin(u)u0 (tan(u))0 = sec2(u)u0 (cot(u))0 = ¡ csc2(u)u0 (sec(u))0 = sec(u) tan(u)u0 (csc(u))0 = ¡ csc(u) cot(u)u0 u0 ¡u0 (sin¡1(u))0 = p (cos¡1(u))0 = p 1 ¡ u2 1 ¡ u2 u0 ¡u0 (tan¡1(u))0 = (cot¡1(u))0 = 1 + u2 1 + u2 u0 ¡u0 (sec¡1(u))0 = p (csc¡1(u))0 = p u u2 ¡ 1 u u2 ¡ 1 General integration de¯nitions and methods: R 1- Inde¯nite integral f(x)dx = F (x) + C means F 0(x) = f(x), F is antiderivative of f R 2- De¯nite integral b f(x)dx = F (b) ¡ F (a) is area under y = f(x) from x = a to x = b R a R R 3- Linearity (af + bg)dx = a fdx + b gdx R R 4a- Integration by parts fg0dx = fg ¡ f 0gdx R R 4b- Integration by parts udv = uv ¡ vdu R R 5a- Inde¯nite integration by substitution f(g(x))g0(x)dx = f(u)du when u = g(x) R R b 0 g(b) 5b- De¯nite integration by substitution a f(g(x))g (x)dx = g(a) f(u)du when u = g(x) 6- Integration by partial fraction decomposition 7- Integration by trigonometric substitution, reduction, circulation, etc 8- Study Chapter 7 of calculus text (Stewart's) for more detail Some basic integration formulas: Z Z un+1 du undu = + C, n 6= ¡1 = ln(u) + C n + 1 u Z Z au eudu = eu + C audu = + C ln a Z Z cos(u)du = sin(u) + C sin(u)du = ¡ cos(u) + C Z Z sec2(u)du = tan(u) + C csc2(u)du = ¡ cot(u) + C Z Z sec(u) tan(u)du = sec(u) + C csc(u) cot(u)du = ¡ csc(u) + C Z Z tan(u)du = ln j sec(u)j + C cot(u)du = ¡ ln j csc(u)j + C Z Z 1 1 ³u´ 1 ³u´ du = tan¡1 + C p du = sin¡1 + C u2 + a2 a a a2 ¡ u2 a Z ¯ ¯ Z 1 1 ¯u ¡ a¯ 1 p du = ln ¯ ¯ + C p du = ln ju + u2 § a2j + C u2 ¡ a2 2a ¯u + a¯ u2 § a2 Z Z sec(u)du = ln j sec(u) + tan(u)j + C csc(u)du = ln j csc(u) ¡ cot(u)j + C Z 1 sec3(u)du = (sec(u) tan(u) + ln j sec(u) + tan(u)j) + C 2 Z 1 csc3(u)du = (¡ csc(u) cot(u) + ln j csc(u) ¡ cot(u)j) + C 2.