Chapter 5: Q 11. (page 495)
Fill in the blanks to complete each of the following integration formulas.
Short Answer
The obtained result is.
Chapter 5: Q 11. (page 495)
Fill in the blanks to complete each of the following integration formulas.
The obtained result is.
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Get started for freeFor each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.
Consider the integral from the reading at the beginning of the section.
(a) Use the inverse trigonometric substitution to solve this integral.
(b) Use the trigonometric substitution to solve the integral.
(c) Compare and contrast the two methods used in parts (a) and (b).
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.
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
Explain why and are essentially the same integral after a change of variables.
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