Due: Wed Jan :01 PM MST. Question Instructions Read today's Notes and Learning Goals

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1 31 asic: Partial Fractions I ( ) Due: Wed Jan :01 PM MST Question Instructions Read today's Notes and Learning Goals 1. Question Details fa15 Partial Frac Intro 0 [ ] Consider the following two quadratic polynomials: x 2 x C 5x 2 6x 11 These two polynomials are equal if their coefficients (the constant factors in front of each term) are equal. Which of the following system of equations would you use to find the values of, and C that make the above polynomials equal? = 5 = 6 C = 11 = 5x 2 = 6x 1 C = 11x 0 = x 2 = x 1 C = x 0 = 5 = 6 C = 11

2 2. Question Details fa15 Partial Frac Intro 02 [ ] Consider the following two quadratic polynomials: ( )x 2 ( C)x C 7x 2 2x 3 These two polynomials are equal if their coefficients are equal. Which of the following system of equations would you use to find the values of, and C that make the above polynomials equal? = x 2 C = x 1 C = x 0 = 7 C = 2 C = 3 = 7 = 2 C = 3 = 7 C = 2 C = 3 3. Question Details sp15 Partial Frac Intro 1 [ ] Consider the following two quadratic polynomials: These two polynomials are equal if their coefficients are equal. Which of the following system of equations would you use to find the values of, and C that make the above polynomials equal? ()x 2 (C 2)x ( C) x 2 4 = 1 C 2 = 0 C = 4 = 4 C 2 = 0 C = 1 = 1 C = 4 = x 2 C 2 = x 1 C = x 0

3 4. Question Details sp15 Partial Frac Intro 2 [ ] Consider the following two linear polynomials: ( )x (23) 10 a. These two polynomials are equal if their coefficients are equal. Which of the following system of equations would you use to find the values of and that make the above polynomials equal? = = 10 ( )x = = 0 = = 0 = = 0 b. Solve the above system of equations for and. = = c. Check your answer by substituting your values for and into the top polynomial ( )x (23), and making sure it is the same as the bottom polynomial 10.

4 5. Question Details sp15 Partial Frac Intro 3 [ ] Partial fraction decomposition starts with a single rational expression: 5x 10 (1 x)(x 4) ssume that this fraction is the result of adding the following two fractions: 1 x x 4 a. Find the least common denominator and combine the two terms into a single 1 x x 4 fraction. Select the correct result. ( )x (4 ) (1 x)(x 4) 5x 10 (1 x)(x 4) 5 ( )x (4 ) (1 x)(x 4) b. Compare the numerator of your result from part (1) to the numerator in the original rational 5x 10 expression, and set up a system of equations to find the values of and (1 x)(x 4) that make the numerators equal. Select the correct system. = 10 4 = 5 = 10 4 = 5 = 5 4 = 10 = 5 4 = 10 c. Solve the system of equations to find the values of and. This allows you to split the original fraction into two fractions: 5x 10 = (1 x)(x 4) 1 x x 4

5 6. Question Details sp15 Partial Frac Intro 4 [ ] Partial fraction decomposition starts with a single rational expression: 3x 2 x(x 1) ssume that this fraction is the result of adding the following two fractions: x x 1 a. Find the least common denominator and combine the two terms into a single x x 1 fraction. b. Compare the numerator of your result in part (1) to the numerator in the original rational 3x 2 expression. Then, set up and solve a system of equations to find the values of and x(x 1) that make the numerators equal. (If you like, you can try an alternative method for setting up a system of equations to solve for and, demonstrated here.) c. Use the values of and you found in part (2) to write the original fraction as a sum of two fractions: 3x 2 x(x 1) =

6 7. Question Details sp16mod Partial Frac Intro 5 [ ] Partial fraction decomposition starts with a single rational expression: ssume that this fraction is the result of adding the following three fractions: The following steps give an alternative method to setup a system of equations for,, and C. 4x 22 (x 1)(x 2)(x 3) x 1 x 2 C x 3 Click here to see a video of this process. a. Write your partial fraction decomposition as an equation as follows 4x 22 (x 1)(x 2)(x 3) = x 1 x 2 C x 3 Multiply both sides of the equation by the least common denominator to get the new equation 4x 22 = (x 2)(x 3) (x 1)(x 3) C(x 1)(x 2) b. Using the nice value x = 1 in the above equation will simplify the equation so it only involves the unknown constant. What is the resulting equation found by using x = 1? c. Solve the equation found in part b for. = d. Use other nice values of x to find the remaining two unknown constants, and C. e. Use the values of, and C to write the original fraction as the sum of three fractions: 4x 22 (x 1)(x 2)(x 3) = 8. Question Details sp15 Partial Frac 1 [ ] Evaluate the following indefinite integral using the method of Partial Fractions. 1 x(x 1) dx a. ssume that the integrand can be written as the sum of the following two fractions: x x 1 Set up a system of equations and solve for the constants and. b. Use the partial fraction decomposition of the integrand to rewrite the integral. c. Use the partial fraction decomposition to find the antiderivative of the original rational expression. Use K for the constant of integration. 1 x (x 1) dx =

7 9. Question Details sp15 Partial Frac 1 [ ] Evaluate the following indefinite integral using the method of Partial Fractions. 10x (x 1)(2x 3) dx a. ssume that the integrand can be written as the sum of the following two fractions: x 1 2x 3 Set up a system of equations and solve for the constants and. b. Use the partial fraction decomposition of the integrand to rewrite the integral. c. Use the partial fraction decomposition to find the antiderivative of the original rational expression. Use K for the constant of integration. 10x (x 1)(2x 3) dx = 10. Question Details sp15 Partial Frac 2 [ ] Evaluate the following indefinite integral using the method of Partial Fractions. 5 x 2 (x 5) dx 1. ssume that the integrand can be written as the sum of the following three fractions: x x 2 C x 5 Set up a system of equations and solve for the constants, and C. 2. Use the partial fraction decomposition of the integrand to rewrite the integral. 3. Use the partial fraction decomposition to find the antiderivative of the original rational expression. Use K for the constant of integration. 5 x 2 (x 5) dx =

8 11. Question Details sp16mod Partial Frac Random given form [ ] Evaluate the following indefinite integral using the method of Partial Fractions. 4x 2 (x 1)(4x 1) dx a. ssume that the integrand can be written as the sum of the following two fractions: x 1 4x 1 Set up a system of equations and solve for the constants and. b. Use the partial fraction decomposition of the integrand to rewrite the integral. c. Use the partial fraction decomposition to find the antiderivative of the original rational expression. Use K for the constant of integration. 4x 2 (x 1)(4x 1) dx = Practice nother Version to get another random partial fractions problem. ssignment Details

MATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By Dr. Mohammed Ramidh

MATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By TECHNIQUES OF INTEGRATION OVERVIEW The Fundamental Theorem connects antiderivatives and the definite integral. Evaluating the indefinite integral,