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 Table of integrals · 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 Table of integrals · 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 Table of integrals · 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 Table of integrals · 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 Table of integrals · Level 2
\( \displaystyle\int_{0}^{\dfrac{\pi}{8}} \arctan 2 x d x \)
6 Table of integrals · Level 2
\( \displaystyle\int_{0}^{2} x^2 \sqrt{4 - x^2} d x \)
7 Table of integrals · Level 2
\( \int \dfrac{\cos x}{\sin^2 x} d x \)
8 Table of integrals · Level 2
\( \int \dfrac{e^x}{1 - e^{2 x}} d x \)
9 Table of integrals · Level 2
\( \int \dfrac{\sqrt{9 x^2 + 4}}{x^2} d x \)
10 Table of integrals · Level 2
\( \int \dfrac{\sqrt{2 y^2 - 3}}{y^2} d y \)
11 Table of integrals · Level 2
\( \displaystyle\int_{0}^{\pi} \cos^6 \theta d \theta \)
12 Table of integrals · Level 2
\( \int x \sqrt{2 + x^4} d x \)
13 Table of integrals · Level 2
\( \int \dfrac{\arctan \sqrt{x}}{\sqrt{x}} d x \)
14 Table of integrals · Level 2
\( \displaystyle\int_{0}^{\pi} x^3 \sin x d x \)
15 Table of integrals · Level 3
\( \int \dfrac{coth\left(\dfrac{1}{y}\right)}{y^2} d y \)
16 Table of integrals · Level 3
\( \int \dfrac{e^{3 t}}{\sqrt{e^{2 t} - 1}} d t \)
17 Table of integrals · Level 3
\( \int y \sqrt{6 + 4 y - 4 y^2} d y \)
18 Table of integrals · Level 2
\( \int \dfrac{d x}{2 x^3 - 3 x^2} \)
19 Table of integrals · Level 3
\( \int \sin^2 x \cos x \ln(\sin x) d x \)
20 Table of integrals · Level 3
\( \int \dfrac{\sin 2 \theta}{\sqrt{5 - \sin \theta}} d \theta \)
21 Table of integrals · Level 2
\( \int \dfrac{e^x}{3 - e^{2 x}} d x \)
22 Table of integrals · Level 3
\( \displaystyle\int_{-2}^2 x^3 \sqrt{4 x^2 - x^4} d x \)
23 Table of integrals · Level 3
\( \int \sec^5 x d x \)
24 Table of integrals · Level 3
\( \int x^3 \arcsin(x^2) d x \)
25 Table of integrals · Level 3
\( \int \dfrac{\sqrt{4 + (\ln x)^2}}{x} d x \)
26 Table of integrals · Level 2
\( \displaystyle\int_{0}^{1} x^4 e^{-x} d x \)
27 Table of integrals · Level 3
\( \int \dfrac{\cos^{-1}(x^{-2})}{x^3} d x \)
28 Table of integrals · Level 2
\( \int \dfrac{d x}{\sqrt{1 - e^{2 x}}} \)
29 Table of integrals · Level 3
\( \int \sqrt{e^{2 x} - 1} d x \)
30 Table of integrals · Level 2
\( \int e^t \sin(\alpha t - 3) d t \)
31 Table of integrals · Level 3
\( \int \dfrac{x^4 d x}{\sqrt{x^{10} - 2}} \)
32 Table of integrals · Level 3
\( \int \dfrac{\sec^2 \theta \tan^2 \theta}{\sqrt{9 - \tan^2 \theta}} d \theta \)
33 Volume application · 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 Volume application · 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 Verify formula · Level 3
Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution \(t = a + b u\).
36 Verify formula · Level 3
Verify Formula 31 (a) by differentiation and (b) by substituting \(u = a \sin \theta\).
37 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS evaluation · 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 CAS domain analysis · 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 CAS with substitution · 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 Example - Volume using table · 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 Example - Table with substitution · Level 3
Use the Table of Integrals to find \(\int \dfrac{x^2}{\sqrt{5 - 4 x^2}} d x\).
49 Example - Reduction formula from table · Level 3
Use the Table of Integrals to evaluate \(\int x^3 \sin x d x\).
50 Example - Completing the square with table · Level 4
Use the Table of Integrals to find \(\int x \sqrt{x^2 + 2 x + 4} d x\).
51 Example - CAS comparison · Level 3
Use a computer algebra system to find \(\int x \sqrt{x^2 + 2 x + 4} d x\).
52 Example - Hand vs CAS · Level 2
Use a CAS to evaluate \(\int x(x^2 + 5)^8 d x\).
53 Example - CAS trig comparison · Level 3
Use a CAS to find \(\int \sin^5 x \cos^2 x d x\).

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