Stewart 8th Section 7.2: Trigonometric Integrals

79 questions

0 / 79
Stewart 8th Section 7.2: Trigonometric Integrals 0/79
1 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin^2 x \cos^3 x d x\).
2 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin^3 \theta \cos^4 \theta d \theta\).
3 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^7 \theta \cos^5 \theta d \theta\).
4 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^5 x d x\).
5 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin^5(2t) \cos^2(2t) d t\).
6 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int t \cos^5(t^2) d t\).
7 Integration and Accumulation of Change · Level 1
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos^2 \theta d \theta\).
8 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{2 \pi} \sin^2\left(\dfrac{1}{3} \theta\right) d \theta\).
9 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\pi} \cos^4(2t) d t\).
10 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{\pi} \sin^2 t \cos^4 t d t\).
11 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^2 x \cos^2 x d x\).
12 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} (2 - \sin \theta)^2 d \theta\).
13 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sqrt{\cos \theta} \sin^3 \theta d \theta\).
14 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \dfrac{\sin^2\left(\dfrac{1}{t}\right)}{t^2} d t\).
15 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \cot x \cos^2 x d x\).
16 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^2 x \cos^3 x d x\).
17 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin^2 x \sin 2x d x\).
18 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin x \cos\left(\dfrac{1}{2} x\right) d x\).
19 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int t \sin^2 t d t\).
20 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int x \sin^3 x d x\).
21 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan x \sec^3 x d x\).
22 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^2 \theta \sec^4 \theta d \theta\).
23 Integration and Accumulation of Change · Level 1
Evaluate the integral: \(\int \tan^2 x d x\).
24 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int (\tan^2 x + \tan^4 x) d x\).
25 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^4 x \sec^6 x d x\).
26 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \sec^6 \theta \tan^6 \theta d \theta\).
27 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^3 x \sec x d x\).
28 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int \tan^5 x \sec^3 x d x\).
29 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^3 x \sec^6 x d x\).
30 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^4 t d t\).
31 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \tan^5 x d x\).
32 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int \tan^2 x \sec x d x\).
33 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int x \sec x \tan x d x\).
34 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \dfrac{\sin \phi}{\cos^3 \phi} d \phi\).
35 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{6}}^{\dfrac{\pi}{2}} \cot^2 x d x\).
36 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \cot^3 x d x\).
37 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \cot^5 \phi \csc^3 \phi d \phi\).
38 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \csc^4 \theta \cot^4 \theta d \theta\).
39 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int \csc x d x\).
40 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\displaystyle\int_{\dfrac{\pi}{6}}^{\dfrac{\pi}{3}} \csc^3 x d x\).
41 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin 8x \cos 5x d x\).
42 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin 2 \theta \sin 6 \theta d \theta\).
43 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos 5t \cos 10t d t\).
44 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \sin x \sec^5 x d x\).
45 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{6}} \sqrt{1 + \cos 2x} d x\).
46 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \sqrt{1 - \cos 4 \theta} d \theta\).
47 Integration and Accumulation of Change · Level 2
Evaluate the integral: \(\int \dfrac{1 - \tan^2 x}{\sec^2 x} d x\).
48 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int \dfrac{d x}{\cos x - 1}\).
49 Integration and Accumulation of Change · Level 3
Evaluate the integral: \(\int x \tan^2 x d x\).
50 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Applications of Integration · Level 2
Find the average value of the function \(f(x) = \sin^2 x \cos^3 x\) on the interval \([-\pi, \pi]\).
56 Differentiation: Definition and Fundamental Properties · 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 Applications of Integration · 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 Applications of Integration · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Applications of Integration · 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 Contextual Applications of Differentiation · 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 Contextual Applications of Differentiation · 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 Differential Equations · 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 Differential Equations · 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 Differential Equations · 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 Infinite Sequences and 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 Integration and Accumulation of Change · Level 2
Evaluate \(\int \cos^3 x d x\).
72 Integration and Accumulation of Change · Level 2
Find \(\int \sin^5 x \cos^2 x d x\).
question image
73 Integration and Accumulation of Change · Level 2
Evaluate \(\displaystyle\int_{0}^{\pi} \sin^2 x d x\).
question image
74 Integration and Accumulation of Change · Level 3
Find \(\int \sin^4 x d x\).
75 Integration and Accumulation of Change · Level 2
Evaluate \(\int \tan^6 x \sec^4 x d x\).
76 Integration and Accumulation of Change · Level 3
Find \(\int \tan^5 \theta \sec^7 \theta d \theta\).
77 Integration and Accumulation of Change · Level 2
Find \(\int \tan^3 x d x\).
78 Integration and Accumulation of Change · Level 3
Find \(\int \sec^3 x d x\).
79 Integration and Accumulation of Change · Level 2
Evaluate \(\int \sin 4x \cos 5x d x\).

Answered: 0 / 79