Chapter 5: Q 20. (page 495)
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,
Chapter 5: Q 20. (page 495)
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,
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Get started for freeComplete 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.
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.
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 if an integrand involves the expression
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: is 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 is of the form
(g) True or False: The partial-fraction decomposition of is of the form .
(h) True or False: Every quadratic function can be written in the form
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