MATH 27 LEC 2.pdf - MATH 27 LECTURE GUIDE UNIT 2 TECHNIQUES...

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MATH 27 Lecture Guide UNIT 2 (IMSP,UPLB) 1 MATH 27 LECTURE GUIDE UNIT 2. TECHNIQUES OF INTEGRATION This unit will focus on the different techniques of integration. Some integrand requires a specific approach to be able to evaluate them. So you need to pay attention to the given integrand so that you will know what techniques to be used. But this does not mean that the simple substitution technique that you learned in the previous unit will no longer work here. In fact, that will always be your first option before applying the other techniques. Our goals for this unit are as follows. By the end of the unit you should be able to perform integration by parts; use trigonometric substitution to evaluate some integral forms; decompose rational functions to partial fractions; use proper substitute to evaluate some integral forms; determine whether an improper integral is convergent or divergent; and determine and execute the proper technique in evaluating integrals . __________________________ REMINDER: REVIEW on integral forms in UNIT 1. These will be the basis of the other solvable integral forms in this unit. Also, review the derivatives for solving differentials in case substitution will be used in solving integrals. 2.1 Integration by Parts In this section, we will study one of the most important techniques of integration called integration by parts . This technique is applicable to the integrand involving products of algebraic and transcendental functions or in some cases, when the integrand is a product of transcendental functions. Here is the forulation: Let u and v be functions of x . Recall the product rule for differentiation, u D v v D u v u D x x x , equivalently it can be expresses as du v dv u v u d . Taking the integrals of both side, yields du v dv u v u d du v dv u v u du v v u dv u MUST REMEMBER!!! Integration by parts (IBP). An integral form can be expressed as which is, in turn, equal to . Once and are determined, solve from , and from . Then, solve the resulting form. MATH 27 Lecture Guide UNIT 2 (IMSP,UPLB) 2 Some helpful tips: 1. Try to let be a function whose derivative is a function simpler than Then will be the remaining factors of the integrand. Note that will always include the of the original integrand. 2. Try to let be the most complicated portion of the integrand that fits a basic integration rule. Then your will be the remaining factor(s) of the integrand. Illustration 1 . Use IBP to evaluate cos x xdx . Solution: Since the derivative of will result to a simple expression, we can choose and so that and Thus, cos sin sin x xdx x x xdx sin ( cos ) sin cos x x x C x x x C   Verify that the answer is correct by showing that ( ) . Illustration 2 . Use IBP to evaluate Arcsin xdx .

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