Lecture 5 (Completing the Square)

Lecture 5 (Completing the Square) - a variation there of....

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Wednesday, January 10 Lecture 5 : Still More Integration Methods : Completing the square and rational functions. ( Not explicitly in your text ) After having practiced the problems associated to the concepts of this lecture the student should be able to : Solve integrals containing expressions such as ax 2 + bx + c by completing the square. 5.1 Introduction Integrals that involve a quadratic expression such as ax 2 + bx + c where neither a nor b is 0 can often be evaluated by first completing the square and then applying the trig substitution method. This approach may be used whenever we cannot obtain the integral of an expression containing ax 2 + bx + c directly. 5.2 Example Find Solution: Completing the square of 3 2 x x 2 to obtain
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5.3 Example Find Solution: We begin by completing the square of the expression 5 – 4 x – 2 x 2 . Completing the square sets up the integrand so that it can be solved by trig substitution or
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Unformatted text preview: a variation there of. 5.4 Example Solve Solution: We begin by completing the square of the quadratic expression: Integration can now be done by trig substitution or a variation there of. From this we obtain Integrating rational functions 5.5 Definition Any function which is a quotient of polynomials is called a rational function . 5.6 When to perform long division If an integrand is a rational function in which the degree of the denominator is less than or equal to the degree of the numerator we normally try to express it at a sum of two simpler rational functions by performing " long division ". 5.6.1 Example Find Solution: 5.6.2 Example Find Solution: In the next lecture we consider integration of rational functions where the degree of numerator is less than the degree of the denominator....
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This note was uploaded on 04/01/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.

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Lecture 5 (Completing the Square) - a variation there of....

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