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Defining improper integrals: Fill in the blanks, using limits and proper definite integrals to express each of the following types of improper integrals.

If f is a continuous on -,, then for any real number c,

-fxdx=___+___

Short Answer

Expert verified

limd-dcfxdx+limecefxdx

Step by step solution

01

Step 1. Given information.

Consider the given integral,

-fxdx

02

Step 2. Defining improper integrals.

Since the given function is continuous on the given interval -,. As c is any real number then break the integral interval to rewrite it.

-fxdx=-cfxdx+cfxdx=limd-dcfxdx+limecefxdx

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2+6x2

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

13-x2dx

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

Explain why it makes sense to try the trigonometric substitution x=secuif an integrand involves the expression x21

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

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