MATH-27-PREP-Integrals-with-weekly-exer.pdf - 1|MATH 27...

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1|MATH 27 PREPARATORY MODULE II| IMSP,UPLB MATH 27 BASIC CALCULUS Preparatory Modules Module II. INTEGRALS Sections 2.1 Basic antidifferentiation 2.2 Substitution technique 2.3 Definite integral 2.4 Area of a region For Week 3. Exercise 2A. Sections 2.1 and 2.2 For Week 4. Exercise 2B. Sections 2.3 and 2.4 _______________________________________ 2.1 BASIC ANTIDIFFERENTIATION A function is called an antiderivative of the function if . A function has several antiderivatives. If is a particular antiderivative of on an interval , then every antiderivative of is of the form , where is an arbitrary constant. Notation: = The arbitrary constant is called constant of integration. Antidifferentiation is the process of finding all the antiderivatives of a given function. The expression is read as “the antiderivative of . An antiderivative is also called indefinite integral . The symbol is called the integral sign and is referred to as the integrand. REMEMBER THESE! Basic rules of antidifferentiation (Power rule) If is any rational number, then , where . CAUTION!!! No immediate rule for antiderivative of product nor of quotient. Examples. 1. To compute , simply apply the Power Rule for antidifferentiation.   dx x f   dx x f   x f   x f dx x 2
2|MATH 27 PREPARATORY MODULE II| IMSP,UPLB Hence, . 2. The Power Rule can also be used to compute antiderivatives of functions with rational exponents. So, . 3. To compute for , simply take the antiderivative of each term. Hence, . 4. There is no quotient rule for antiderivatives. Some quotients can be written in the form where Power Rule is applicable. So, . 5. There is no product rule for antiderivatives. Whenever possible, first take the product of the expressions. Then, compute for the term-by-term antiderivatives. Hence, . ____________________________________________ 2.2 SUBSTITUTION TECHNIQUE Simple substitution techniques can be used to compute some integral forms. This is best used in solving integrals of some products or quotients.

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