AC Integrating Rational Functions

A. $\int (\frac{A}{x} + \frac{B}{x - 5} + \frac{C}{x + 9}) d x$
B. $\int (\frac{A}{x} + \frac{B x + C}{x - 10} + \frac{D x + E}{x + 5}) d x$
C. There is no partial fraction decomposition because the denominator does not factor.
D. $\int (\frac{A}{x - 10} + \frac{B}{x - 9} + \frac{C}{x + 5}) d x$
E. There is no partial fraction decomposition yet because there is cancellation.
F. There is no partial fraction decomposition yet because long division must be done first.
G. $\int (\frac{A}{x} + \frac{B}{x - 10} + \frac{C}{x + 5}) d x$
H. $\int (\frac{A x + B}{x} + \frac{C x + D}{x - 5} + \frac{E x + F}{x + 9}) d x$

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4.

What is the correct form of the partial fraction decomposition for the following integral?

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\int \frac{10 (x^{8} + 6)}{(x - 7) (x^{2} - 1)^{2} (x^{2} + 7)^{2}} d x

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A. There is no partial fraction decomposition because the denominator does not factor.
B. $\int (\frac{A}{x - 7} + \frac{B}{x + 1} + \frac{C}{(x + 1)^{2}} + \frac{D}{x - 1} + \frac{E}{(x - 1)^{2}} + \frac{F x + G}{x^{2} + 7}) d x$
C. $\int (\frac{A}{x - 7} + \frac{B x + C}{(x^{2} - 1)^{2}} + \frac{D x + E}{(x^{2} + 7)^{2}}) d x$
D. $\int (\frac{A}{x - 7} + \frac{B x + C}{x^{2} - 1} + \frac{C x + D}{(x^{2} - 1)^{2}} + \frac{E x + F}{x^{2} + 7} + \frac{G x + H}{(x^{2} + 7)^{2}}) d x$
E. $\int (\frac{A}{x - 7} + \frac{B}{(x^{2} - 1)^{2}} + \frac{C}{(x^{2} + 7)^{2}}) d x$
F. There is no partial fraction decomposition yet because long division must be done first.
G. There is no partial fraction decomposition yet because there is cancellation.

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5.

Consider the indefinite integral

\int \frac{6 x^{3} + 2 x^{2} - 49 x - 21}{x^{2} - 9} d x

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Then the integrand decomposes into the form

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a x + b + \frac{c}{x - 3} + \frac{d}{x + 3}

where

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a

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b

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c

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d

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Integrating term by term, we obtain that

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\int \frac{6 x^{3} + 2 x^{2} - 49 x - 21}{x^{2} - 9} d x =

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+ C

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6.

Evaluate the integral

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\int \frac{10}{(x + a) (x + b)} d x

for the cases where

a = b

and where

a \neq b .

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Note: For the case where

a = b,

use only

a

in your answer. Also, use an upper-case "C" for the constant of integration.

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a = b :

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a \neq b :

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7.

Consider the integral

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\int \frac{x^{21} - 3 x^{14} + 6 x^{7} - 16}{{(x^{3} - 6 x^{2} + 5 x)}^{3} {(x^{4} - 625)}^{2}} d x

Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand.

A_{1}, A_{2}, A_{3} \dots

and

B_{1}, B_{2}, B_{3}, \dots

denote constants.

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You must get all of the answers correct to receive credit.

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8.

Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same.

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\int \frac{2 x}{x^{2} - 4} d x

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First, rewrite this with partial fractions:

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\int \frac{2 x}{x^{2} - 4} d x = \int

d x + \int

d x =

+ C .

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(Note that you should not include the $+ C$ in your entered answer, as it has been provided at the end of the expression.)

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Next, use the substitution

w = x^{2} - 4

to find the integral:

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\int \frac{2 x}{x^{2} - 4} d x = \int

d w =

+ C =

+ C .

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(For the second answer blank, give your antiderivative in terms of the variable $w .$ Again, note that you should not include the $+ C$ in your answer.)

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