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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11,13,17,19
![### Integral Problems for Practice
#### Evaluate the following integrals:
1. \(\int_{0}^{\pi/2} \sin^2 x \cos^2 x \, dx\)
2. \(\int_{0}^{\pi/2} (2 - \sin \theta)^2 \, d\theta\)
3. \(\int \sqrt{\cos \theta} \sin^3 \theta \, d\theta\)
4. \(\int \left(1 + \sqrt[3]{\sin t} \right) \cos^3 t \, dt\)
5. \(\int \sin x \sec^5 x \, dx\)
6. \(\int \csc^5 \theta \cos^3 \theta \, d\theta\)
7. \(\int \cot x \cos^2 x \, dx\)
8. \(\int \tan^2 x \cos^3 x \, dx\)
9. \(\int \sin^2 x \sin 2x \, dx\)
10. \(\int \sin x \cos \left(\frac{1}{2} x \right) \, dx\)
#### Notes for Understanding:
- Each integral requires applying appropriate integration techniques, such as substitution, integration by parts, or trigonometric identities.
- Pay attention to the bounds of integration for definite integrals.
- For indefinite integrals, remember to add the constant of integration \(C\).
Please solve these integrals carefully and verify your results with standard antiderivative rules and formulas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4dc4e677-4da4-4b74-b6dc-2448ffced4fa%2Fea47d541-0f39-4d7f-81e7-05d4ef8fc525%2Fmy0oj8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Problems for Practice
#### Evaluate the following integrals:
1. \(\int_{0}^{\pi/2} \sin^2 x \cos^2 x \, dx\)
2. \(\int_{0}^{\pi/2} (2 - \sin \theta)^2 \, d\theta\)
3. \(\int \sqrt{\cos \theta} \sin^3 \theta \, d\theta\)
4. \(\int \left(1 + \sqrt[3]{\sin t} \right) \cos^3 t \, dt\)
5. \(\int \sin x \sec^5 x \, dx\)
6. \(\int \csc^5 \theta \cos^3 \theta \, d\theta\)
7. \(\int \cot x \cos^2 x \, dx\)
8. \(\int \tan^2 x \cos^3 x \, dx\)
9. \(\int \sin^2 x \sin 2x \, dx\)
10. \(\int \sin x \cos \left(\frac{1}{2} x \right) \, dx\)
#### Notes for Understanding:
- Each integral requires applying appropriate integration techniques, such as substitution, integration by parts, or trigonometric identities.
- Pay attention to the bounds of integration for definite integrals.
- For indefinite integrals, remember to add the constant of integration \(C\).
Please solve these integrals carefully and verify your results with standard antiderivative rules and formulas.
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