Jan. 14 Rational Functions- Completing the square

Jan. 14 Rational Functions- Completing the square -...

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Friday January 14 Lecture 5 : Still More Integration Methods : Completing the square and rational functions. ( Refers to Section 7.4 of your text ) Expectations: 1. Solve integrals containing expressions such as ax 2 + bx + c by completing the square. 2. Solve integrals containing rational functions. 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. 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 a variation there of. 5.4 Example – Solve
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Unformatted text preview: 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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Jan. 14 Rational Functions- Completing the square -...

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