Integrated Two Unrelated Functions | University of California, Berkeley

University:
University of California, Berkeley
Course:
MATH 206 | Functional Analysis
Academic year:

2019
Views:

37

Pages:

6
Author:

Clint W.

When integrating two unrelated functions, integration by parts is used. ∫[u dv] = uv - ∫[v du] The function representing U is determined by descending through the following list, with the mnemonic device "LIPET" for "Long Island PET": Logarithms Inverse Trigonometric functions Polynomials (i.e., algebraic functions) e^x (exponential functions) Trigonometric functions It may be necessary to repeat the process. Ex. on next page Tabular integration can also be used when one function can be differentiated in some degree to zero and the other function is easy to integrate. Ex. ∫x³cos2x dx x³ → 3x² → 6x → 6 → 0 cos2x → ½sin2x → -¼cos2x → ¼sin2x → -1/8cos2x Study Rules for Differentiation/Integration Integration by Parts ∫[u dv] = uv - ∫[v du] LIPET (u in ascending order) Logarithmic functions Inverse trigonometric functions Polynomials (algebraic functions) e^x functions Trigonometric functions Tabular Integration Differentiate one function in some degree to zero and the other function is easy to integrate. Variable Separable Treat dy/dx as a fraction Partial Fractions Unrestricted Growth dy/dt = ky → y = Ce^(kt) Newton's Law of Cooling dy/dt = -k(y-A) → y = A + Ce^(-kt) A: temperature of surrounding C: initial temperature - surrounding temperature Rate of change is greatest between y=A and y=m Logistic Growth dy/dt = ky(m-y)/m → y = m/(1+Ce^(-kt)) m: carrying capacity Rate of increase is greatest between y=0 and y=m/2 L'Hôpital's Rule For the limits of fractions in the indeterminate form 0/0 or ∞/∞: If lim(x→a)f(x) = 0 or ±∞ and lim(x→a)g(x) = 0 or ±∞, and f'(a) and g'(a) both exist, then lim(x→a)f(x)/g(x) = lim(x→a)f'(x)/g'(x) Special Cases of L'Hôpital's Rule For the limits of an indeterminate form 0^∞, ∞^0, 1^∞: lim(x→a)f(x)^g(x) = e^lim(x→a)g(x)ln[f(x)] Rates of Growth From greatest order to least: x!, a^x, x^a, x, ln(x), log_a(x), where a > 1 Let f(x) and g(x) be positive for x sufficiently large. - f grows faster than g if lim(x→∞)f(x)/g(x) = ∞, or lim(x→∞)g(x)/f(x) = 0. - f and g grow at the same rate if lim(x→∞)f(x)/g(x) = L, where L is a nonzero constant. Transitivity of the rate of growth: When x approaches infinity, if f grows at the same rate as g, and g grows at the same rate as h, then f grows at the same rate as h. f = o(g): f is of lower order than g (lim(x→∞)f(x)/g(x) = 0) f = O(g): f is at most (of lower or equal order) the order of g (lim(x→∞)f(x)/g(x) is finite) Improper Integrals Definite integrals with a "problem spot" are expressed as integrals with the "problem spots" as limits in the interval. If the "problem spot" occurs in the middle of the interval, one-sided limits are used. Converge/Diverge Converge: function totals up to a finite number Diverge: function does not "total up" to a finite number; infinity, does not exist