Integration Techniques: Substitution, Parts, Trig, Fractions, and Improper Integrals, Study notes of Calculus

Download Integration Techniques: Substitution, Parts, Trig, Fractions, and Improper Integrals and more Study notes Calculus in PDF only on Docsity! Math 102-001, Spring 2008, Stolz Some recommendations on using integration techniques 1. Substitution • A substitution u = g(x) works best, if the derivative g′(x) appears as part of the integrand (but there are exceptions). • A substitution can be used to use an integrand to a form which can be treated with other methods (e.g. one of the forms suitable for trigonometric substitution). Example: ∫ x5 √ x3 − 1 dx. 2. Integration by Parts • If an integrand has two factors, which of them should one integrate and which one differentiate when using integration by part? As a rule: First choose log functions to differentiate. If no log functions are present, choose power functions. Examples: In ∫ xex dx choose u = x, dv = exdx, while in ∫ x ln x dx choose u = ln x, dv = xdx (it would not be good to choose u = x, dv = ln xdx here). • In integrals of the form ∫ ex sin x dx choose u = sin x and dv = exdx, do two integrations by part and then solve for the unknown integral. • Example: ∫ xex sin x dx. Start by finding ∫ ex sin x dx and then choose u = x, dv = ex sin x dx to integrate the original integral by parts. 3. Trigonometric Integrals • Integrals of the form ∫ sinm x cosn x dx: If m or n are odd, then use sinx + cos2 x = 1 on one of the odd-powered terms and then substitute u = sin x or u = cos x. Example: ∫ sin5 x cos4 dx = ∫ (1 − cos2 x)2 sin x cos3 x dx. Substitute u = cos x. If m and n are both odd, then reduce the integrand to odd powered sines or cosines by using half-angle identities. 1