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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### Evaluating the Definite Integral
**Problem Statement:**
Evaluate the definite integral:
\[
\int_{0}^{2} \frac{x}{\sqrt{1 + 4x}} \, dx
\]
**Instructions:**
To solve this definite integral, we need to integrate the function \( \frac{x}{\sqrt{1 + 4x}} \) with respect to \( x \) over the interval from 0 to 2.
**Step-by-Step Solution:**
1. **Substitution Method:**
- Let \( u = 1 + 4x \).
- Then, differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = 4 \quad \Rightarrow \quad dx = \frac{du}{4} \]
2. **Change the Limits of Integration:**
- When \( x = 0 \), \( u = 1 \).
- When \( x = 2 \), \( u = 1 + 4(2) = 9 \).
3. **Rewrite the Integral in terms of \( u \):**
\[
\int_{1}^{9} \frac{u - 1}{4\sqrt{u}} \, du
\]
- Simplify the integrand:
\[
\frac{1}{4} \int_{1}^{9} \left( \frac{u}{\sqrt{u}} - \frac{1}{\sqrt{u}} \right) du = \frac{1}{4} \int_{1}^{9} \left( u^{1/2} - u^{-1/2} \right) du
\]
4. **Integrate:**
\[
\frac{1}{4} \left[ \frac{2}{3} u^{3/2} - 2u^{1/2} \right]_{1}^{9}
\]
- Find the antiderivative:
\[
\frac{1}{4} \left[ \frac{2}{3} u^{3/2} - 2u^{1/2} \right]
\]
5. **Evaluate the Definite Integral:**
- Apply the limits of integration:
\[
\frac{1}{4} \left\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2F1a441d0e-a7d8-4754-8042-ff25d5613f25%2Fsz0nuhl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Evaluating the Definite Integral
**Problem Statement:**
Evaluate the definite integral:
\[
\int_{0}^{2} \frac{x}{\sqrt{1 + 4x}} \, dx
\]
**Instructions:**
To solve this definite integral, we need to integrate the function \( \frac{x}{\sqrt{1 + 4x}} \) with respect to \( x \) over the interval from 0 to 2.
**Step-by-Step Solution:**
1. **Substitution Method:**
- Let \( u = 1 + 4x \).
- Then, differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = 4 \quad \Rightarrow \quad dx = \frac{du}{4} \]
2. **Change the Limits of Integration:**
- When \( x = 0 \), \( u = 1 \).
- When \( x = 2 \), \( u = 1 + 4(2) = 9 \).
3. **Rewrite the Integral in terms of \( u \):**
\[
\int_{1}^{9} \frac{u - 1}{4\sqrt{u}} \, du
\]
- Simplify the integrand:
\[
\frac{1}{4} \int_{1}^{9} \left( \frac{u}{\sqrt{u}} - \frac{1}{\sqrt{u}} \right) du = \frac{1}{4} \int_{1}^{9} \left( u^{1/2} - u^{-1/2} \right) du
\]
4. **Integrate:**
\[
\frac{1}{4} \left[ \frac{2}{3} u^{3/2} - 2u^{1/2} \right]_{1}^{9}
\]
- Find the antiderivative:
\[
\frac{1}{4} \left[ \frac{2}{3} u^{3/2} - 2u^{1/2} \right]
\]
5. **Evaluate the Definite Integral:**
- Apply the limits of integration:
\[
\frac{1}{4} \left\
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