Integration Techniques: Study Guide for Substitution, Parts, Trig Integrals, and Trig Sub , Study notes of Analytical Geometry and Calculus

Download Integration Techniques: Study Guide for Substitution, Parts, Trig Integrals, and Trig Sub and more Study notes Analytical Geometry and Calculus in PDF only on Docsity! Techniques of Integration Study Guide Charlie Egedy April 3, 2009 Make certain that you know the basic integrals, those listed as 1-16 in Rogawski’s Table of Integrals (back part of the book), along with the integral ∫ sec3(x)dx = 12 [sec(x) tan(x) + ln | sec(x) + tan(x)|]. You will find integral 16 to particularly useful in solving partial fraction decomposition problems with minimal effort. 1 Substitution • Think of substitution as reversing the chain rule for differentiation. • Find a substitution that simplifies the rule for a function by replacing a repeated element or a particularly complicated element by a single variable. • Compute the differential of the new variable in terms of the old viariable. If u = g(x), then du = ( dg dx ) dx. • If the integral is a definite integral, compute limits of integration in terms of the new variable. • Make the substitution, noting that every occurrence of the old variable must be replaced. • Either complete the process of integration or employ another technique as necessary. 2 Integration by Parts • Think of integration by parts as reversing the product rule for differentiation. • The basic theory states that ∫ udv = uv − ∫ vdu. • The entire integrand must be incorporated into u and v. Keep in mind the maxim: Do No Harm! In general, you want the processes of differentiation and integration to remove difficulties without introducing new ones. • If the integrand is the product of a polynomial and either a trig function or an exponential, then set u equal to the polynomial and dv equal to the remainder of the integrand. • If the integrand is the product of an exponential and a trig function, then in general it does not matter how you make the assignment; however, if you set u equal to the trig function and another round of integration by parts is required, then in the second round, u must again be set equal to the trig function. We make the analogous requirement if u is initially chosen to be the exponential. • If the integrand is the product of a polynomial and a logarithm, then set u equal to the logarithm. 1