Stewart 8th Section 7.6: Integration Using Tables and Computer Algebra Systems

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Stewart 8th Section 7.6: Integration Using Tables and Computer Algebra Systems 0/53
1 Integration and Accumulation of Change · Level 2
Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos 5 x \cos 2 x d x\); entry 80
2 Integration and Accumulation of Change · Level 2
Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. \(\displaystyle\int_{0}^{1} \sqrt{x - x^2} d x\); entry 113
3 Integration and Accumulation of Change · Level 2
Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. \(\displaystyle\int_{1}^{2} \sqrt{4 x^2 - 3} d x\); entry 39
4 Integration and Accumulation of Change · Level 2
Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. \(\displaystyle\int_{0}^{1} \tan^3 \left(\pi \dfrac{x}{6}\right) d x\); entry 69
5 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{\dfrac{\pi}{8}} \arctan 2 x d x \)
6 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{2} x^2 \sqrt{4 - x^2} d x \)
7 Integration and Accumulation of Change · Level 2
\( \int \dfrac{\cos x}{\sin^2 x} d x \)
8 Integration and Accumulation of Change · Level 2
\( \int \dfrac{e^x}{1 - e^{2 x}} d x \)
9 Integration and Accumulation of Change · Level 2
\( \int \dfrac{\sqrt{9 x^2 + 4}}{x^2} d x \)
10 Integration and Accumulation of Change · Level 2
\( \int \dfrac{\sqrt{2 y^2 - 3}}{y^2} d y \)
11 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{\pi} \cos^6 \theta d \theta \)
12 Integration and Accumulation of Change · Level 2
\( \int x \sqrt{2 + x^4} d x \)
13 Integration and Accumulation of Change · Level 2
\( \int \dfrac{\arctan \sqrt{x}}{\sqrt{x}} d x \)
14 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{\pi} x^3 \sin x d x \)
15 Integration and Accumulation of Change · Level 3
\( \int \dfrac{coth\left(\dfrac{1}{y}\right)}{y^2} d y \)
16 Integration and Accumulation of Change · Level 3
\( \int \dfrac{e^{3 t}}{\sqrt{e^{2 t} - 1}} d t \)
17 Integration and Accumulation of Change · Level 3
\( \int y \sqrt{6 + 4 y - 4 y^2} d y \)
18 Integration and Accumulation of Change · Level 2
\( \int \dfrac{d x}{2 x^3 - 3 x^2} \)
19 Integration and Accumulation of Change · Level 3
\( \int \sin^2 x \cos x \ln(\sin x) d x \)
20 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sin 2 \theta}{\sqrt{5 - \sin \theta}} d \theta \)
21 Integration and Accumulation of Change · Level 2
\( \int \dfrac{e^x}{3 - e^{2 x}} d x \)
22 Integration and Accumulation of Change · Level 3
\( \displaystyle\int_{-2}^2 x^3 \sqrt{4 x^2 - x^4} d x \)
23 Integration and Accumulation of Change · Level 3
\( \int \sec^5 x d x \)
24 Integration and Accumulation of Change · Level 3
\( \int x^3 \arcsin(x^2) d x \)
25 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sqrt{4 + (\ln x)^2}}{x} d x \)
26 Integration and Accumulation of Change · Level 2
\( \displaystyle\int_{0}^{1} x^4 e^{-x} d x \)
27 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\cos^{-1}(x^{-2})}{x^3} d x \)
28 Integration and Accumulation of Change · Level 2
\( \int \dfrac{d x}{\sqrt{1 - e^{2 x}}} \)
29 Integration and Accumulation of Change · Level 3
\( \int \sqrt{e^{2 x} - 1} d x \)
30 Integration and Accumulation of Change · Level 2
\( \int e^t \sin(\alpha t - 3) d t \)
31 Integration and Accumulation of Change · Level 3
\( \int \dfrac{x^4 d x}{\sqrt{x^{10} - 2}} \)
32 Integration and Accumulation of Change · Level 3
\( \int \dfrac{\sec^2 \theta \tan^2 \theta}{\sqrt{9 - \tan^2 \theta}} d \theta \)
33 Applications of Integration · Level 3
The region under the curve \(y = \sin^2 x\) from 0 to \(\pi\) is rotated about the \(x\)-axis. Find the volume of the resulting solid.
34 Applications of Integration · Level 3
Find the volume of the solid obtained when the region under the curve \(y = \arcsin x\), \(x \geq 0\), is rotated about the \(y\)-axis.
35 Integration and Accumulation of Change · Level 3
Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution \(t = a + b u\).
36 Integration and Accumulation of Change · Level 3
Verify Formula 31 (a) by differentiation and (b) by substituting \(u = a \sin \theta\).
37 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \sec^4 x d x\)
38 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \csc^5 x d x\)
39 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int x^2 \sqrt{x^2 + 4} d x\)
40 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \dfrac{d x}{e^x (3 e^x + 2)}\)
41 Integration and Accumulation of Change · Level 2
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \cos^4 x d x\)
42 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int x^2 \sqrt{1 - x^2} d x\)
43 Integration and Accumulation of Change · Level 2
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \tan^5 x d x\)
44 Integration and Accumulation of Change · Level 3
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \dfrac{1}{\sqrt{1 + \sqrt[3]{x}}} d x\)
45 Limits and Continuity · Level 4
(a) Use the table of integrals to evaluate \(F(x) = \int f(x) d x\), where \(f(x) = \dfrac{1}{x \sqrt{1 - x^2}}\). What is the domain of \(f\) and \(F\)?
(b) Use a CAS to evaluate \(F(x)\). What is the domain of the function \(F\) that the CAS produces? Is there a discrepancy between this domain and the domain of the function \(F\) that you found in part (a)?

Enter your answer directly below each part above.

46 Integration and Accumulation of Change · Level 4
Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate \(\int (1 + \ln x) \sqrt{1 + (x \ln x)^2} d x\) with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the CAS can evaluate.
47 Applications of Integration · Level 3
The region bounded by the curves \(y = \arctan x\), \(y = 0\), and \(x = 1\) is rotated about the \(y\)-axis. Find the volume of the resulting solid.
48 Integration and Accumulation of Change · Level 3
Use the Table of Integrals to find \(\int \dfrac{x^2}{\sqrt{5 - 4 x^2}} d x\).
49 Integration and Accumulation of Change · Level 3
Use the Table of Integrals to evaluate \(\int x^3 \sin x d x\).
50 Integration and Accumulation of Change · Level 4
Use the Table of Integrals to find \(\int x \sqrt{x^2 + 2 x + 4} d x\).
51 Applications of Integration · Level 3
Use a computer algebra system to find \(\int x \sqrt{x^2 + 2 x + 4} d x\).
52 Integration and Accumulation of Change · Level 2
Use a CAS to evaluate \(\int x(x^2 + 5)^8 d x\).
53 Applications of Integration · Level 3
Use a CAS to find \(\int \sin^5 x \cos^2 x d x\).

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