Stewart 8th Section 7.2: Trigonometric Integrals

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Stewart 8th Section 7.2: Trigonometric Integrals 0/79
1 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin^2 x \cos^3 x d x\).
2 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin^3 \theta \cos^4 \theta d \theta\).
3 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^7 \theta \cos^5 \theta d \theta\).
4 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^5 x d x\).
5 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin^5(2t) \cos^2(2t) d t\).
6 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int t \cos^5(t^2) d t\).
7 Exercise - Trigonometric Integrals · Level 1
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\).
8 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{2 \pi} \sin^2\left(\dfrac{1}{3} \theta\right) d \theta\).
9 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\pi} \cos^4(2t) d t\).
10 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{\pi} \sin^2 t \cos^4 t d t\).
11 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^2 x \cos^2 x d x\).
12 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} (2 - \sin \theta)^2 d \theta\).
13 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sqrt{\cos \theta} \sin^3 \theta d \theta\).
14 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \dfrac{\sin^2\left(\dfrac{1}{t}\right)}{t^2} d t\).
15 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \cot x \cos^2 x d x\).
16 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^2 x \cos^3 x d x\).
17 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin^2 x \sin 2x d x\).
18 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin x \cos\left(\dfrac{1}{2} x\right) d x\).
19 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int t \sin^2 t d t\).
20 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int x \sin^3 x d x\).
21 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan x \sec^3 x d x\).
22 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^2 \theta \sec^4 \theta d \theta\).
23 Exercise - Trigonometric Integrals · Level 1
Evaluate the integral: \(\int \tan^2 x d x\).
24 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int (\tan^2 x + \tan^4 x) d x\).
25 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^4 x \sec^6 x d x\).
26 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \sec^6 \theta \tan^6 \theta d \theta\).
27 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^3 x \sec x d x\).
28 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int \tan^5 x \sec^3 x d x\).
29 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^3 x \sec^6 x d x\).
30 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^4 t d t\).
31 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \tan^5 x d x\).
32 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int \tan^2 x \sec x d x\).
33 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int x \sec x \tan x d x\).
34 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \dfrac{\sin \phi}{\cos^3 \phi} d \phi\).
35 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{6}}^{\dfrac{\pi}{2}} \cot^2 x d x\).
36 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \cot^3 x d x\).
37 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \cot^5 \phi \csc^3 \phi d \phi\).
38 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \csc^4 \theta \cot^4 \theta d \theta\).
39 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int \csc x d x\).
40 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{6}}^{\dfrac{\pi}{3}} \csc^3 x d x\).
41 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin 8x \cos 5x d x\).
42 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin 2 \theta \sin 6 \theta d \theta\).
43 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos 5t \cos 10t d t\).
44 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \sin x \sec^5 x d x\).
45 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{6}} \sqrt{1 + \cos 2x} d x\).
46 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \sqrt{1 - \cos 4 \theta} d \theta\).
47 Exercise - Trigonometric Integrals · Level 2
Evaluate the integral: \(\int \dfrac{1 - \tan^2 x}{\sec^2 x} d x\).
48 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int \dfrac{d x}{\cos x - 1}\).
49 Exercise - Trigonometric Integrals · Level 3
Evaluate the integral: \(\int x \tan^2 x d x\).
50 Exercise - Reduction Relationship · Level 3
If \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^6 x \sec x d x = I\), express the value of \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^8 x \sec x d x\) in terms of \(I\).
51 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C = 0\)): \(\int x \sin^2(x^2) d x\).
52 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C = 0\)): \(\int \sin^5 x \cos^3 x d x\).
53 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C = 0\)): \(\int \sin 3x \sin 6x d x\).
54 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C = 0\)): \(\int \sec^4\left(\dfrac{1}{2} x\right) d x\).
55 Exercise - Average Value · Level 2
Find the average value of the function \(f(x) = \sin^2 x \cos^3 x\) on the interval \([-\pi, \pi]\).
56 Exercise - Multiple Methods · Level 2
Evaluate \(\int \sin x \cos x d x\) by four methods: (a) the substitution \(u = \cos x\), (b) the substitution \(u = \sin x\), (c) the identity \(\sin 2x = 2 \sin x \cos x\), (d) integration by parts. Explain the different appearances of the answers.
57 Exercise - Area Between Curves · Level 2
Find the area of the region bounded by the given curves: \(y = \sin^2 x\), \(y = \sin^3 x\), \(0 \leq x \leq \pi\).
58 Exercise - Area Between Curves · Level 2
Find the area of the region bounded by the given curves: \(y = \tan x\), \(y = \tan^2 x\), \(0 \leq x \leq \dfrac{\pi}{4}\).
59 Exercise - Graphical Estimation · Level 2
Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct: \(\displaystyle\int_{0}^{2 \pi} \cos^3 x d x\).
60 Exercise - Graphical Estimation · Level 2
Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct: \(\displaystyle\int_{0}^{2} \sin 2 \pi x \cos 5 \pi x d x\).
61 Exercise - Volume of Revolution · Level 3
Find the volume obtained by rotating the region bounded by the curves about the given axis: \(y = \sin x\), \(y = 0\), \(\dfrac{\pi}{2} \leq x \leq \pi\); about the \(x\)-axis.
62 Exercise - Volume of Revolution · Level 3
Find the volume obtained by rotating the region bounded by the curves about the given axis: \(y = \sin^2 x\), \(y = 0\), \(0 \leq x \leq \pi\); about the \(x\)-axis.
63 Exercise - Volume of Revolution · Level 3
Find the volume obtained by rotating the region bounded by the curves about the given axis: \(y = \sin x\), \(y = \cos x\), \(0 \leq x \leq \dfrac{\pi}{4}\); about \(y = 1\).
64 Exercise - Volume of Revolution · Level 3
Find the volume obtained by rotating the region bounded by the curves about the given axis: \(y = \sec x\), \(y = \cos x\), \(0 \leq x \leq \dfrac{\pi}{3}\); about \(y = -1\).
65 Exercise - Particle Motion · Level 2
A particle moves on a straight line with velocity function \(v(t) = \sin \omega t \cos^2 \omega t\). Find its position function \(s = f(t)\) if \(f(0) = 0\).
66 Exercise - Electricity Application · Level 3
Household electricity is supplied in the form of alternating current that varies from 155 V to \(-155\) V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the equation \(E(t) = 155 \sin(120 \pi t)\) where \(t\) is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of \([E(t)]^2\) over one cycle. (a) Calculate the RMS voltage of household current. (b) Many electric stoves require an RMS voltage of 220
V. Find the corresponding amplitude \(A\) needed for the voltage \(E(t) = A \sin(120 \pi t)\).
67 Exercise - Orthogonality Proof · Level 3
Prove the formula, where \(m\) and \(n\) are positive integers: \(\displaystyle\int_{-\pi}^\pi \sin m x \cos n x d x = 0\).
68 Exercise - Orthogonality Proof · Level 3
Prove the formula, where \(m\) and \(n\) are positive integers: \(\displaystyle\int_{-\pi}^\pi \sin m x \sin n x d x = \begin{cases} 0 & \quad \text{if } m \neq n \\ \pi & \quad \text{if } m = n \end{cases}\).
69 Exercise - Orthogonality Proof · Level 3
Prove the formula, where \(m\) and \(n\) are positive integers: \(\displaystyle\int_{-\pi}^\pi \cos m x \cos n x d x = \begin{cases} 0 & \quad \text{if } m \neq n \\ \pi & \quad \text{if } m = n \end{cases}\).
70 Exercise - Fourier Series · Level 3
A finite Fourier series is given by the sum \(f(x) = \displaystyle\sum_{n=1}^N a_n \sin n x = a_1 \sin x + a_2 \sin 2x + \ldots + a_N \sin N x\). Show that the \(m\)th coefficient \(a_m\) is given by the formula \(a_m = \dfrac{1}{\pi} \displaystyle\int_{-\pi}^\pi f(x) \sin m x d x\).
71 Example - Odd Power of Cosine · Level 2
Evaluate \(\int \cos^3 x d x\).
72 Example - Odd Power of Sine · Level 2
Find \(\int \sin^5 x \cos^2 x d x\).
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73 Example - Half-Angle Identity · Level 2
Evaluate \(\displaystyle\int_{0}^{\pi} \sin^2 x d x\).
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74 Example - Power Reduction · Level 3
Find \(\int \sin^4 x d x\).
75 Example - Tangent and Secant (Even Secant) · Level 2
Evaluate \(\int \tan^6 x \sec^4 x d x\).
76 Example - Tangent and Secant (Odd Tangent) · Level 3
Find \(\int \tan^5 \theta \sec^7 \theta d \theta\).
77 Example - Tangent Only · Level 2
Find \(\int \tan^3 x d x\).
78 Example - Odd Power of Secant · Level 3
Find \(\int \sec^3 x d x\).
79 Example - Product to Sum Identity · Level 2
Evaluate \(\int \sin 4x \cos 5x d x\).

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