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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![**Evaluate the following definite integrals:**
a) \[ \int_{0}^{\pi} (5e^x + 3\sin x) \, dx \]
b) \[ \int_{-3}^{-1} \left( \frac{1}{x^2} + \frac{1}{x} \right) \, dx \]
c) \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{8}{1 + x^2} \, dx \]
**Explanation:**
This section contains three definite integrals. Each problem requires evaluating an integral over a specified interval.
- **Part (a):** The integral involves an exponential function \( e^x \) and a trigonometric function \( \sin x \). The integral is taken from 0 to \(\pi\).
- **Part (b):** This involves two functions: \(\frac{1}{x^2}\) and \(\frac{1}{x}\), integrated over the interval from -3 to -1.
- **Part (c):** This integral features a rational expression with \( 1 + x^2 \) in the denominator. The limits of integration are from \(\frac{1}{\sqrt{3}}\) to \(\sqrt{3}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8bddee-32fd-433e-8085-8fa50d25b5ad%2Fdced9840-821f-4830-a050-1e71b5d729aa%2F2f3qey_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the following definite integrals:**
a) \[ \int_{0}^{\pi} (5e^x + 3\sin x) \, dx \]
b) \[ \int_{-3}^{-1} \left( \frac{1}{x^2} + \frac{1}{x} \right) \, dx \]
c) \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{8}{1 + x^2} \, dx \]
**Explanation:**
This section contains three definite integrals. Each problem requires evaluating an integral over a specified interval.
- **Part (a):** The integral involves an exponential function \( e^x \) and a trigonometric function \( \sin x \). The integral is taken from 0 to \(\pi\).
- **Part (b):** This involves two functions: \(\frac{1}{x^2}\) and \(\frac{1}{x}\), integrated over the interval from -3 to -1.
- **Part (c):** This integral features a rational expression with \( 1 + x^2 \) in the denominator. The limits of integration are from \(\frac{1}{\sqrt{3}}\) to \(\sqrt{3}\).
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