Stewart 8th Section 7.4: Integration of Rational Functions by Partial Fractions

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Stewart 8th Section 7.4: Integration of Rational Functions by Partial Fractions 0/84
1 Integration and Accumulation of Change · Level 1
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{4 + x}{(1 + 2 x)(3 - x)}\)
(b) \(\dfrac{1 - x}{x^3 + x^4}\)

Enter your answer directly below each part above.

2 Integration and Accumulation of Change · Level 1
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{x - 6}{x^2 + x - 6}\)
(b) \(\dfrac{x^2}{x^2 + x + 6}\)

Enter your answer directly below each part above.

3 Integration and Accumulation of Change · Level 1
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{1}{x^2 + x^4}\)
(b) \(\dfrac{x^3 + 1}{x^3 - 3 x^2 + 2 x}\)

Enter your answer directly below each part above.

4 Integration and Accumulation of Change · Level 2
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{x^4 - 2 x^3 + x^2 + 2 x - 1}{x^2 - 2 x + 1}\)
(b) \(\dfrac{x^2 - 1}{x^3 + x^2 + x}\)

Enter your answer directly below each part above.

5 Integration and Accumulation of Change · Level 2
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{x^6}{x^2 - 4}\)
(b) \(\dfrac{x^4}{(x^2 - x + 1)(x^2 + 2)^2}\)

Enter your answer directly below each part above.

6 Integration and Accumulation of Change · Level 2
Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients.
(a) \(\dfrac{t^6 + 1}{t^6 + t^3}\)
(b) \(\dfrac{x^5 + 1}{(x^2 - x)(x^4 + 2 x^2 + 1)}\)

Enter your answer directly below each part above.

7 Integration and Accumulation of Change · Level 2
\( \int \dfrac{x^4}{x - 1} d x \)
8 Integration and Accumulation of Change · Level 1
\( \int \dfrac{3 t - 2}{t + 1} d t \)
9 Integration and Accumulation of Change · Level 2
\( \int \dfrac{5 x + 1}{(2 x + 1)(x - 1)} d x \)
10 Integration and Accumulation of Change · Level 2
\( \int \dfrac{y}{(y + 4)(2 y - 1)} d y \)
11 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{1} \dfrac{2}{2 x^2 + 3 x + 1} d x \)
12 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{1} \dfrac{x - 4}{x^2 - 5 x + 6} d x \)
13 Integration and Accumulation of Change · Level 2
\( \int \dfrac{a x}{x^2 - b x} d x \)
14 Integration and Accumulation of Change · Level 2
\( \int \dfrac{1}{(x + a)(x + b)} d x \)
15 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{-1}^0 \dfrac{x^3 - 4 x + 1}{x^2 - 3 x + 2} d x \)
16 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{1}^{2} \dfrac{x^3 + 4 x^2 + x - 1}{x^3 + x^2} d x \)
17 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{1}^{2} \dfrac{4 y^2 - 7 y - 12}{y(y + 2)(y - 3)} d y \)
18 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{1}^{2} \dfrac{3 x^2 + 6 x + 2}{x^2 + 3 x + 2} d x \)
19 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{x^2 + x + 1}{(x + 1)^2(x + 2)} d x \)
20 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{2}^{3} \dfrac{x(3 - 5 x)}{(3 x - 1)(x - 1)^2} d x \)
21 Integration and Accumulation of Change · Level 3
\( \int \dfrac{d t}{(t^2 - 1)^2} \)
22 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^4 + 9 x^2 + x + 2}{x^2 + 9} d x \)
23 Integration and Accumulation of Change · Level 3
\( \int \dfrac{10}{(x - 1)(x^2 + 9)} d x \)
24 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^2 - x + 6}{x^3 + 3 x} d x \)
25 Integration and Accumulation of Change · Level 3
\( \int \dfrac{4 x}{x^3 + x^2 + x + 1} d x \)
26 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^2 + x + 1}{(x^2 + 1)^2} d x \)
27 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^3 + 4 x + 3}{x^4 + 5 x^2 + 4} d x \)
28 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^3 + 6 x - 2}{x^4 + 6 x^2} d x \)
29 Integration and Accumulation of Change · Level 2
\( \int \dfrac{x + 4}{x^2 + 2 x + 5} d x \)
30 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^3 - 2 x^2 + 2 x - 5}{x^4 + 4 x^2 + 3} d x \)
31 Integration and Accumulation of Change · Level 3
\( \int \dfrac{1}{x^3 - 1} d x \)
32 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{x}{x^2 + 4 x + 13} d x \)
33 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{x^3 + 2 x}{x^4 + 4 x^2 + 3} d x \)
34 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^5 + x - 1}{x^3 + 1} d x \)
35 Integration and Accumulation of Change · Level 4
\( \int \dfrac{5 x^4 + 7 x^2 + x + 2}{x(x^2 + 1)^2} d x \)
36 Integration and Accumulation of Change · Level 4
\( \int \dfrac{x^4 + 3 x^2 + 1}{x^5 + 5 x^3 + 5 x} d x \)
37 Integration and Accumulation of Change · Level 4
\( \int \dfrac{x^2 - 3 x + 7}{(x^2 - 4 x + 6)^2} d x \)
38 Integration and Accumulation of Change · Level 4
\( \int \dfrac{x^3 + 2 x^2 + 3 x - 2}{(x^2 + 2 x + 2)^2} d x \)
39 Integration and Accumulation of Change · Level 2
\( \int \dfrac{d x}{x \sqrt{x - 1}} \)
40 Integration and Accumulation of Change · Level 3
\( \int \dfrac{d x}{2 \sqrt{x + 3} + x} \)
41 Integration and Accumulation of Change · Level 3
\( \int \dfrac{d x}{x^2 + x \sqrt{x}} \)
42 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{1}{1 + \sqrt[3]{x}} d x \)
43 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^3}{\sqrt[3]{x^2 + 1}} d x \)
44 Integration and Accumulation of Change · Level 3
\( \int \dfrac{d x}{(1 + \sqrt{x})^2} \)
45 Integration and Accumulation of Change · Level 3
\(\int \dfrac{1}{\sqrt{x} - \sqrt[3]{x}} d x\) [Hint: Substitute \(u = \sqrt[6]{x}\).]
46 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sqrt{1 + \sqrt{x}}}{x} d x \)
47 Integration and Accumulation of Change · Level 3
\( \int \dfrac{e^{2 x}}{e^{2 x} + 3 e^x + 2} d x \)
48 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sin x}{\cos^2 x - 3 \cos x} d x \)
49 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sec^2 t}{\tan^2 t + 3 \tan t + 2} d t \)
50 Integration and Accumulation of Change · Level 4
\( \int \dfrac{e^x}{(e^x - 2)(e^{2 x} + 1)} d x \)
51 Integration and Accumulation of Change · Level 2
\( \int \dfrac{d x}{1 + e^x} \)
52 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\cosh t}{\sinh^2 t + \sinh^4 t} d t \)
53 Integration and Accumulation of Change · Level 4
Use integration by parts, together with the techniques of this section, to evaluate the integral. \(\int \ln(x^2 - x + 2) d x\)
54 Integration and Accumulation of Change · Level 3
Use integration by parts, together with the techniques of this section, to evaluate the integral. \(\int x \tan^{-1} x d x\)
55 Analytical Applications of Differentiation · Level 2
Use a graph of \(f(x) = 1/(x^2 - 2 x - 3)\) to decide whether \(\displaystyle\int_{0}^{2} f(x) d x\) is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.
56 Limits and Continuity · Level 3
Evaluate \(\int \dfrac{1}{x^2 + k} d x\) by considering several cases for the constant \(k\).
57 Integration and Accumulation of Change · Level 2
Evaluate the integral by completing the square and using Formula 6. \(\int \dfrac{d x}{x^2 - 2 x}\)
58 Integration and Accumulation of Change · Level 3
Evaluate the integral by completing the square and using Formula 6. \(\int \dfrac{2 x + 1}{4 x^2 + 12 x - 7} d x\)
59 Integration and Accumulation of Change · Level 4
The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution \(t = \tan\left(\dfrac{x}{2}\right)\) will convert any rational function of \(\sin x\) and \(\cos x\) into an ordinary rational function of \(t\).
(a) If \(t = \tan\left(\dfrac{x}{2}\right)\), \(-\pi < x < \pi\), sketch a right triangle or use trigonometric identities to show that \(\cos\left(\dfrac{x}{2}\right) = \dfrac{1}{\sqrt{1 + t^2}}\) and \(\sin\left(\dfrac{x}{2}\right) = \dfrac{t}{\sqrt{1 + t^2}}\)
(b) Show that \(\cos x = \dfrac{1 - t^2}{1 + t^2}\) and \(\sin x = \dfrac{2 t}{1 + t^2}\)
(c) Show that \(d x = \dfrac{2}{1 + t^2} d t\)

Enter your answer directly below each part above.

60 Integration and Accumulation of Change · Level 3
Use the substitution in Exercise 59 to transform the integrand into a rational function of \(t\) and then evaluate the integral. \(\int \dfrac{d x}{1 - \cos x}\)
61 Integration and Accumulation of Change · Level 3
Use the substitution in Exercise 59 to transform the integrand into a rational function of \(t\) and then evaluate the integral. \(\int \dfrac{1}{3 \sin x - 4 \cos x} d x\)
62 Integration and Accumulation of Change · Level 4
Use the substitution in Exercise 59 to transform the integrand into a rational function of \(t\) and then evaluate the integral. \(\displaystyle\int_{\dfrac{\pi}{3}}^{\dfrac{\pi}{2}} \dfrac{1}{1 + \sin x - \cos x} d x\)
63 Integration and Accumulation of Change · Level 4
Use the substitution in Exercise 59 to transform the integrand into a rational function of \(t\) and then evaluate the integral. \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{\sin 2 x}{2 + \cos x} d x\)
64 Integration and Accumulation of Change · Level 3
Find the area of the region under the given curve from 1 to 2. \(y = \dfrac{1}{x^3 + x}\)
65 Integration and Accumulation of Change · Level 3
Find the area of the region under the given curve from 1 to 2. \(y = \dfrac{x^2 + 1}{3 x - x^2}\)
66 Applications of Integration · Level 3
Find the volume of the resulting solid if the region under the curve \(y = 1/(x^2 + 3 x + 2)\) from \(x = 0\) to \(x = 1\) is rotated about (a) the \(x\)-axis and (b) the \(y\)-axis.
67 Differential Equations · Level 4
One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. (The photo shows a screw-worm fly, the first pest effectively eliminated from a region by this method.) Let \(P\) represent the number of female insects in a population and \(S\) the number of sterile males introduced each generation. Let \(r\) be the per capita rate of production of females by females, provided their chosen mate is not sterile. Then the female population is related to time \(t\) by \(t = \int \dfrac{P + S}{P[(r - 1)P - S]} d P\) Suppose an insect population with 10,000 females grows at a rate of \(r = 1.1\) and 900 sterile males are added initially. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation can't be solved explicitly for \(P\).)
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68 Limits and Continuity · Level 3
Factor \(x^4 + 1\) as a difference of squares by first adding and subtracting the same quantity. Use this factorization to evaluate \(\int 1/(x^4 + 1) d x\).
69 Integration and Accumulation of Change · Level 3
(a) Use a computer algebra system to find the partial fraction decomposition of the function \(f(x) = \dfrac{4 x^3 - 27 x^2 + 5 x - 32}{30 x^5 - 13 x^4 + 50 x^3 - 286 x^2 - 299 x - 70}\)
(b) Use part (a) to find \(\int f(x) d x\) (by hand) and compare with the result of using the CAS to integrate \(f\) directly. Comment on any discrepancy.

Enter your answer directly below each part above.

70 Integration and Accumulation of Change · Level 3
(a) Find the partial fraction decomposition of the function \(f(x) = \dfrac{12 x^5 - 7 x^3 - 13 x^2 + 8}{100 x^6 - 80 x^5 + 116 x^4 - 80 x^3 + 41 x^2 - 20 x + 4}\)
(b) Use part (a) to find \(\int f(x) d x\) and graph \(f\) and its indefinite integral on the same screen.
(c) Use the graph of \(f\) to discover the main features of the graph of \(\int f(x) d x\).

Enter your answer directly below each part above.

71 Integration and Accumulation of Change · Level 4
The rational number \(\dfrac{22}{7}\) has been used as an approximation to the number \(\pi\) since the time of Archimedes. Show that \(\displaystyle\int_{0}^{1} \dfrac{x^4 (1 - x)^4}{1 + x^2} d x = \dfrac{22}{7} - \pi\)
72 Integration and Accumulation of Change · Level 4
(a) Use integration by parts to show that, for any positive integer \(n\), \(\int \dfrac{d x}{(x^2 + a^2)^n} d x = \dfrac{x}{2 a^2 (n - 1)(x^2 + a^2)^{n - 1}} + \dfrac{2 n - 3}{2 a^2 (n - 1)} \int \dfrac{d x}{(x^2 + a^2)^{n - 1}}\)
(b) Use part (a) to evaluate \(\int \dfrac{d x}{(x^2 + 1)^2}\) and \(\int \dfrac{d x}{(x^2 + 1)^3}\)

Enter your answer directly below each part above.

73 Limits and Continuity · Level 4
Suppose that \(F\), \(G\), and \(Q\) are polynomials and \(\dfrac{F(x)}{Q(x)} = \dfrac{G(x)}{Q(x)}\) for all \(x\) except when \(Q(x) = 0\). Prove that \(F(x) = G(x)\) for all \(x\). [Hint: Use continuity.]
74 Integration and Accumulation of Change · Level 4
If \(f\) is a quadratic function such that \(f(0) = 1\) and \(\int \dfrac{f(x)}{x^2 (x + 1)^3} d x\) is a rational function, find the value of \(f'(0)\).
75 Integration and Accumulation of Change · Level 5
If \(a \neq 0\) and \(n\) is a positive integer, find the partial fraction decomposition of \(f(x) = \dfrac{1}{x^n (x - a)}\). [Hint: First find the coefficient of \(1/(x - a)\). Then subtract the resulting term and simplify what is left.]
76 Integration and Accumulation of Change · Level 2
Find \(\int \dfrac{x^3 + x}{x - 1} d x\).
77 Integration and Accumulation of Change · Level 3
Evaluate \(\int \dfrac{x^2 + 2 x - 1}{2 x^3 + 3 x^2 - 2 x} d x\).
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78 Integration and Accumulation of Change · Level 2
Find \(\int \dfrac{d x}{x^2 - a^2}\), where \(a \neq 0\).
79 Integration and Accumulation of Change · Level 3
Find \(\int \dfrac{x^4 - 2 x^2 + 4 x + 1}{x^3 - x^2 - x + 1} d x\).
80 Integration and Accumulation of Change · Level 3
Evaluate \(\int \dfrac{2 x^2 - x + 4}{x^3 + 4 x} d x\).
81 Integration and Accumulation of Change · Level 3
Evaluate \(\int \dfrac{4 x^2 - 3 x + 2}{4 x^2 - 4 x + 3} d x\).
82 Integration and Accumulation of Change · Level 2
Write out the form of the partial fraction decomposition of the function \(\dfrac{x^3 + x^2 + 1}{x(x - 1)(x^2 + x + 1)(x^2 + 1)^3}\).
83 Integration and Accumulation of Change · Level 4
Evaluate \(\int \dfrac{1 - x + 2 x^2 - x^3}{x(x^2 + 1)^2} d x\).
84 Integration and Accumulation of Change · Level 3
Evaluate \(\int \dfrac{\sqrt{x + 4}}{x} d x\).

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