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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Question
![**Problem Statement**
Evaluate the integral:
\[
\int_{2}^{4} \frac{4 + u^2}{u^3} \, du
\]
**Solution Explanation**
You're asked to solve the definite integral from 2 to 4 of the function \(\frac{4 + u^2}{u^3}\).
1. **Simplify the Integrand**: Break down the term into simpler fractions:
\[
\frac{4 + u^2}{u^3} = \frac{4}{u^3} + \frac{u^2}{u^3} = 4u^{-3} + u^{-1}
\]
2. **Integrate Term by Term**:
- Integrate \(4u^{-3}\):
\[
\int 4u^{-3} \, du = 4 \cdot \frac{u^{-2}}{-2} = -2u^{-2}
\]
- Integrate \(u^{-1}\):
\[
\int u^{-1} \, du = \ln|u|
\]
3. **Combine the Results**:
The antiderivative of the function is:
\[
-2u^{-2} + \ln|u|
\]
4. **Evaluate the Definite Integral**: Use the limits of integration from 2 to 4.
\[
\left[-2\left(\frac{1}{4^2}\right) + \ln|4|\right] - \left[-2\left(\frac{1}{2^2}\right) + \ln|2|\right]
\]
Simplify this expression to find the value of the definite integral.
**Result**
Calculate the final value to solve the integral completely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2379209f-bd09-4933-933b-b65441272cfd%2F73a28690-5d26-48df-8752-e42e8ce3ea19%2F4p5qumd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Evaluate the integral:
\[
\int_{2}^{4} \frac{4 + u^2}{u^3} \, du
\]
**Solution Explanation**
You're asked to solve the definite integral from 2 to 4 of the function \(\frac{4 + u^2}{u^3}\).
1. **Simplify the Integrand**: Break down the term into simpler fractions:
\[
\frac{4 + u^2}{u^3} = \frac{4}{u^3} + \frac{u^2}{u^3} = 4u^{-3} + u^{-1}
\]
2. **Integrate Term by Term**:
- Integrate \(4u^{-3}\):
\[
\int 4u^{-3} \, du = 4 \cdot \frac{u^{-2}}{-2} = -2u^{-2}
\]
- Integrate \(u^{-1}\):
\[
\int u^{-1} \, du = \ln|u|
\]
3. **Combine the Results**:
The antiderivative of the function is:
\[
-2u^{-2} + \ln|u|
\]
4. **Evaluate the Definite Integral**: Use the limits of integration from 2 to 4.
\[
\left[-2\left(\frac{1}{4^2}\right) + \ln|4|\right] - \left[-2\left(\frac{1}{2^2}\right) + \ln|2|\right]
\]
Simplify this expression to find the value of the definite integral.
**Result**
Calculate the final value to solve the integral completely.
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