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Why do we need to put linear numerators of the form Bix+Ciin every term of a partial-fraction decomposition which involves irreducible quadratics? Think about the example pxqx=1x2+12and figure out what goes wrong if we attempt to make a decomposition of the formC1x2+1+C2x2+12.

Short Answer

Expert verified

The qx=1x2+1, whichis a reducible quadratic then qxis a product of l1xand l2xof two linear functions. So,l1x=x-1,l2x=x-2.

Step by step solution

01

Step 1. Given information

pxqx=1x2+12.

02

Step 2. Let us see some concepts about qx.

If qxis an irreducible quadratic then pxqxis its own partial fractions decomposition, there is nothing further we can decompose.

If qxis a reducible quadratic then qxis a product of l1x.l2xof two linear functions and the obtain a partial function decomposition of the form A1l1x+A2l2x.

The given function is, qx=1x2+1.

So,l1x=x-1,l2x=x-2.

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Most popular questions from this chapter

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

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

โˆซu2du

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

(a)โˆซ4+x2xdx (b)โˆซx4+x2dx

role="math" localid="1648759296940" (c)โˆซx24+x2dx (d)โˆซ16โˆ’x44+x2dx

Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.

f(x)=secโˆ’1โกx

Suppose u(x)=x2. Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" โˆซ-15u2du,โˆซx=-1x=5u2duandโˆซu(-1)u(5)u2du

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