Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Integration Problems for Practice
Below are a series of definite and indefinite integrals to help improve your skills in integration, particularly focusing on trigonometric functions. Try to evaluate each integral and verify your answers.
11. \(\int_{0}^{\pi/2} \sin^2{x} \cos^2{x} \, dx\)
12. \(\int_{0}^{\pi/2} (2 - \sin{\theta})^2 \, d\theta\)
13. \(\int \sqrt{\cos{\theta}} \sin^3{\theta} \, d\theta\)
14. \(\int (1 + \sqrt[3]{\sin{t}}) \cos^3{t} \, dt\)
15. \(\int \sin{x} \sec^5{x} \, dx\)
16. \(\int \csc^5{\theta} \cos^3{\theta} \, d\theta\)
17. \(\int \cot{x} \cos^2{x} \, dx\)
18. \(\int \tan^2{x} \cos^3{x} \, dx\)
19. \(\int \sin^2{x} \sin{2x} \, dx\)
20. \(\int \sin{x} \cos{\left(\frac{1}{2}x\right)} \, dx\)
### Explanation of Notations
- The integral symbol \(\int\) signifies the integration operation.
- The limits of integration for definite integrals are specified next to the integral sign (e.g., \(\int_{0}^{\pi/2}\)).
- \(\sin, \cos, \tan, \cot, \sec, \csc\) represent the standard trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant, respectively.
- \(\theta, x, t\) are the variables of integration.
By solving these integrals, you will reinforce your understanding of integral calculus involving trigonometric functions. Solutions to these integrals often involve a combination of techniques such as substitution, integration by parts, and recognizing standard integral forms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4dc4e677-4da4-4b74-b6dc-2448ffced4fa%2F1311f1ff-29c3-4967-8821-0d411e074c0a%2Fy1lv5dq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integration Problems for Practice
Below are a series of definite and indefinite integrals to help improve your skills in integration, particularly focusing on trigonometric functions. Try to evaluate each integral and verify your answers.
11. \(\int_{0}^{\pi/2} \sin^2{x} \cos^2{x} \, dx\)
12. \(\int_{0}^{\pi/2} (2 - \sin{\theta})^2 \, d\theta\)
13. \(\int \sqrt{\cos{\theta}} \sin^3{\theta} \, d\theta\)
14. \(\int (1 + \sqrt[3]{\sin{t}}) \cos^3{t} \, dt\)
15. \(\int \sin{x} \sec^5{x} \, dx\)
16. \(\int \csc^5{\theta} \cos^3{\theta} \, d\theta\)
17. \(\int \cot{x} \cos^2{x} \, dx\)
18. \(\int \tan^2{x} \cos^3{x} \, dx\)
19. \(\int \sin^2{x} \sin{2x} \, dx\)
20. \(\int \sin{x} \cos{\left(\frac{1}{2}x\right)} \, dx\)
### Explanation of Notations
- The integral symbol \(\int\) signifies the integration operation.
- The limits of integration for definite integrals are specified next to the integral sign (e.g., \(\int_{0}^{\pi/2}\)).
- \(\sin, \cos, \tan, \cot, \sec, \csc\) represent the standard trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant, respectively.
- \(\theta, x, t\) are the variables of integration.
By solving these integrals, you will reinforce your understanding of integral calculus involving trigonometric functions. Solutions to these integrals often involve a combination of techniques such as substitution, integration by parts, and recognizing standard integral forms.
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