Applications of Derivatives

Chapter 4

1IntegrationSubtopics24.1 The definite integral4.2 The fundamental theorem and calculus4.3 Substitution techniques4.4 Integration by parts4.5 Integration by partial fractions4.6 Trigonometric integrals4.7 Trigonometric substitutions

Formula Sheet3

4.1 Indefinite and definite integral4Integration is the inverse of differentiation:

Differentiation is the inverse of integration:

The definite integral from a to b

5Refer Tutorial 4

6More exercises

4.2 The fundamental theorem and calculus7

4.3 Substitution Techniques8With algebraic substitutions, the substitution always made is to let u be equal to f(x) such that f(u) du is a standard integral. It is found that integrals of the forms :

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4.4 Integration by Parts10

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4.5 Integration by Partial Fractions12CASE 1 : If all of the factors of g(x) are distinct linear factors, then the partial fraction decomposition of contains the sum :

Example :

4.5 Integration by Partial Fractions13CASE 2 : If the factors of g(x) are repeated linear factors, then the partial fraction decomposition of contains the sum :

Example :

4.5 Integration by Partial Fractions14CASE 3 : If the factors of g(x)=(ax+b)(ax2+bx+c) is a combination of linear and quadratic factors where the quadratic factor cannot be factorized, then the partial fraction decomposition are as follows :

Example :

4.6 Trigonometric Integrals15Please refer to earlier formula sheet to solve these trigonometric problems

4.7 Trigonometric Substitution16

17End Chapter