WebAssign Lesson 4-1 Partial Fractions (Homework)

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1 Webssign Lesson 4-1 Partial Fractions (Homework) Current Score : / 46 Due : Wednesday, July :00 M MDT Jaimos Skriletz Math 175, section 31, Summer Instructor: Jaimos Skriletz 1. /3 points Partial fractions decomposition starts with a single rational expression 5x 10 (1 x)(x 4) 1. ssume that the fraction was the result of adding two fractions together of the form 5x 10 B = (1 x)(x 4) 1 x x 4 The goal is to find the values of the unknown constants and B. 2. Start with the RHS and combine the two terms into a single term. Select the correct result. (You only get 2 tries.) B 5 (B )x (4B ) (1 x)(x 4) ( B)x (4 B) (1 x)(x 4) 5x 10 (1 x)(x 4) 3. Compare the numerator found in the above step to the original fraction. (stuff) x (stuff) = 5 x 10 (1 x)(x 4) (1 x)(x 4) Select the correct system of equations that arises from comparing the answer of step 1 to the original fraction. (You only get two tries.) B = 10 4B = 5 = B5 4 = B10 = B10 4 = B5 B = 5 4B = 10

2 4. Solve the system of equations to find the values of and B. (Enter in exact values, no decimals.) = B = 5. You finish by separating your original fraction into two fractions 1 x B x 4 2. /3 points Partial fractions decomposition starts with a single rational expression 2x 2 3x 5 (x 1)(x 2 4) 1. ssume that the fraction was the result of adding two fractions together of the form 2x 2 3x 5 Bx = (x 1)(x 2 4) x 1 x 2 4 The goal is to find the values of the unknown constants, B and C. 2. Start with the RHS and combine the two terms into a single term. Select the correct result. (You only get 2 tries.) Bx ( ) x 2 x 3 (4 C)x 2 ( B)x (C B) (x 1)(x 2 4) (C B)x 2 (4 C)x ( B) (x 1)(x 2 4) ( B)x 2 (C B)x (4 C) (x 1)(x 2 4) 3. Compare the numerator found in the above step to the original fraction. (stuff) x 2 (stuff) x (stuff) 2x = 2 3x 5 (x 1)(x 2 4) (x 1)(x 2 4) Select the correct system of equations that arises from comparing the answer of step 1 to the original fraction. (You only get two tries.) = B3 C = B5 4 = C2

3 = B5 C = B2 4 = C3 = B2 C = B3 4 = C5 B = 2 B = C3 4C = 5 4. Solve the system of equations to find the values of, B and C. (Enter in exact values, no decimals.) = B = C = 5. You finish by separating your original fraction into two fractions x 1 Bx x /2 points Partial fractions decomposition starts with a single rational expression x (x 3)(2x 5) 1. ssume that the fraction was the result of adding two fractions together of the form x B = (x 3)(2x 5) x 3 2x 5 2. Find the unknown constants and B. = B =

4 4. /3 points Partial fractions decomposition starts with a single rational expression 5 x (3x 1)(x 1) 2 1. ssume that the fraction was the result of adding two fractions together of the form 5 x B = (3x 1)(x 1) 2 3x 1 x 1 2. find the unknown constants, B and C. C (x 1) 2 = B = C =

5 5. /4 points The following are some of the basic integrals that arise in partial fraction decomposition. Use C for the constant of integration. 1. Find the following indefinite integral: 3 5x 2 2. Find the following indefinte integral: 3. Find the following indefinite integral: (Hint: Use 4 (5 3x) 2 u = x 2 4 = 4(5 3x) 2 substitution.) 3 x x Find the following indefinite integral: Hint: Use the elementary antiderivative 1 = tan 1 (x/a) x 2 a 2 a 5 x 2 9

6 6. /3 points Find the following indefinite integral (antiderivative) 10x (x 1)(2x 3) 1. First find the partial fraction decomposition 10x B = (x 1)(2x 3) x 1 2x 3 Find the unknown constants and B. 2. Then use the partial fraction to split the integral into two terms: (Enter in the two terms found above, do not forget to include.) 3. Last find the antiderivative of each term to find the original antiderivaitve. (Do not forget C.) 10x (x 1)(2x 3) 10x (x 1)(2x 3)

7 7. /3 points Find the following indefinite integral (antiderivative) x 2 2x (x 2)(x 2 4) 1. First find the partial fraction decomposition x 2 2x Bx = (x 2)(x 2 4) x 2 x 2 4 Find the unknown constants, B and C. 2. Then use the partial fraction to split the integral into two terms: (Enter in the two terms found above, do not forget to include.) x 2 2x (x 2)(x 2 4) 3. Last find the antiderivative of each term to find the original antiderivaitve. (Do not forget C.) x 2 2x (x 2)(x 2 4)

8 8. /3 points Find the following indefinite integral (antiderivative) 10 x 2 (x 5) 1. First determine the partial fraction decomposition (You only get two tries.) x 2 B x 5 x x x Bx D x 2 x 5 B C x 2 x 5 B x 5 2. Find the unknown constants and split the fraction apart 10 = x 2 (x 5) 3. Use the partial fraction decomposition to find the antiderivative (Do not forget C.) 10 x 2 (x 5)

9 9. /3 points Find the following indefinite integral (antiderivative) x(x 1) (x 2 4)(x 4) 1. First determine the partial fraction decomposition (You only get two tries.) x B x 2 4 x 4 x 2 x B x 2 4 x 2 4 B C x 2 x 4 C x 4 B x 4 2. Find the unknown constants and split the fraction apart x(x 1) = (x 2 4)(x 4) 3. Use the partial fraction decomposition to find the antiderivative (Do not forget C.) x(x 1) (x 2 4)(x 4) 10. /2 pointsrogac alcet Tutorial. Evaluate the integral. (Remember to use ln u where appropriate.) (4x 1) x 2 9x 20 dditional Materials Tutorial

10 11. /2 pointsrogac alcet Evaluate the integral. (Remember to use ln u where appropriate.) 13 (x 4) 2 (x 1) 12. /2 pointsrogac alcet Evaluate the integral. (Remember to use ln u where appropriate.) 7x 2 (x 1)(x 2 1) 13. /2 pointsrogac alcet Evaluate the integral. (Remember to use ln u where appropriate.) 10 x(x 2 25) 14. /2 pointsrogac alcet Evaluate the integral. 9 x 2 (x 2 25)

11 15. /3 points Find the indefinite integral (antiderivative) of 2x 3 x 2 6x 7 x 2 x 6 1. First use polynomial long division to find the quotient and remainder. 2x 3 x 2 6x 7 = x 2 x 6 x 2 x 6 2. The use partial fraction decomposition to split the remainder (fractional) term apart and rewrite the integral as a sum of terms that have elementary antiderivatives. (Note the integral symbol and are provided for you.) 2x 3 x 2 6x 7 x 2 x 6 3. Find the elementary antiderivative of each term to solve the original problem. (Use C for the constant of integration.) 2x 3 x 2 6x 7 x 2 x /2 points Find the indefinite integral (antiderivative) of x 4 5x x First use polynomial long division to find the quotient and remainder. x 4 5x = x Find the indefinite integral (antiderivative) (Use C for the constant of integration.) x 2 25 x 4 5x x 2 25

12 17. /2 pointsrogac alcet Tutorial. Evaluate using long division first to write f(x) as the sum of a polynomial and a proper rational function. (Remember to use ln u where appropriate.) (x 3 8x 2 5) x 8 dditional Materials CalcClip Tutorial 18. /2 pointsrogac alcet Evaluate the integral. (Remember to use ln u where appropriate.) (x 2 5x 1) x 2 x