Review of differentiation and integration rules from I and II for Ordinary Differential Equations, 3301 General Notation: a, b, m, n, C are non-specific constants, independent of variables e, π are special constants e = 2.71828 ···, π = 3.14159 ··· f, g, u, v, F are functions f n(x) usually means [f(x)]n, but f −1(x) usually means inverse 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 first and second . d d2 d d2 Leibnitz notations for first and second derivatives are and or and dx dx2 dt dt2 Differential of x is shown by dx or ∆x or h f(x + ∆x) − f(x) f 0(x) = lim , of f shows the of the line, rise ∆x→0 ∆x over run, for the function y = f(x) at x

General differentiation 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- (fg)0 = f 0g + fg0 µ ¶ f 0 f 0g − fg0 4- = g g2 5- (f n)0 = nf n−1f 0 6- (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 specifies the first differentiation rule to be used

Function-specific differentiation 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 = √ (cos−1(u))0 = √ 1 − u2 1 − u2 u0 −u0 (tan−1(u))0 = (cot−1(u))0 = 1 + u2 1 + u2 u0 −u0 (sec−1(u))0 = √ (csc−1(u))0 = √ u u2 − 1 u u2 − 1

General integration definitions and methods: R 1- Indefinite f(x)dx = F (x) + C means F 0(x) = f(x), F is of f R 2- Definite 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- fg0dx = fg − f 0gdx R R 4b- Integration by parts udv = uv − vdu R R 5a- Indefinite integration by substitution f(g(x))g0(x)dx = f(u)du when u = g(x) R R b 0 g(b) 5b- Definite 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 | sec(u)| + C cot(u)du = − ln | csc(u)| + C Z Z 1 1 ³u´ 1 ³u´ du = tan−1 + C √ du = sin−1 + C u2 + a2 a a a2 − u2 a Z ¯ ¯ Z 1 1 ¯u − a¯ 1 p du = ln ¯ ¯ + C √ du = ln |u + u2 ± a2| + C u2 − a2 2a ¯u + a¯ u2 ± a2 Z Z sec(u)du = ln | sec(u) + tan(u)| + C csc(u)du = ln | csc(u) − cot(u)| + C Z 1 sec3(u)du = (sec(u) tan(u) + ln | sec(u) + tan(u)|) + C 2 Z 1 csc3(u)du = (− csc(u) cot(u) + ln | csc(u) − cot(u)|) + C 2