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Fill in the blanks to complete each of the following theorem statements:

In a partial-fractions decomposition of a proper rational function p(x)q(x), if q(x)has a linear factor x-cwith multiplicity m, then for some constants A1,A2,...,Am, the sum will include terms of the forms _____.

Short Answer

Expert verified

In a partial-fractions decomposition of a proper rational function p(x)q(x), if q(x)has a linear factor x-cwith multiplicity m, then for some constants A1,A2,...,Am, the sum will include terms of the formsA1x-c+A2x-c2+...+Amx-cm.

Step by step solution

01

Step 1. Given information  

In a partial-fractions decomposition of a proper rational function p(x)q(x), if q(x)has a linear factor x-cwith multiplicity m, then for some constants A1,A2,...,Am, the sum will include terms of the forms _____.

02

Step 2. Filling in the blanks to complete the theorem statements  

In a partial-fractions decomposition of a proper rational function p(x)q(x), if q(x)has a linear factor x-cwith multiplicity m, then for some constants A1,A2,...,Am, the sum will include terms of the forms A1x-c+A2x-c2+...+Amx-cm.

Proper rational functions are decomposed into partial fractions for linear factors with some multiplicity as shown above.

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Most popular questions from this chapter

For each integral in Exercises 5โ€“8, write down three integrals that will have that form after a substitution of variables.

โˆซ1udu

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solvingโˆซ1x2โˆ’4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solvingโˆซ1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingโˆซx2+4โˆ’5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2โˆ’a2.

(f) True or False: Trigonometric substitution doesnโ€™t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Consider the integral โˆซx(x2โˆ’1)2dx.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Solve the integral: โˆซxlnx2dx

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