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
![To evaluate the indefinite integral
\[
\int \left( 3 + \frac{1}{t} \right)^6 \frac{1}{t^2} dt
\]
Use C for the constant of integration.
**Explanation:**
There is a mathematical expression given in the form of an indefinite integral. The integral involves the function \(\left( 3 + \frac{1}{t} \right)^6\) divided by \(t^2\), and the variable of integration is \(t\). The task is to find the integral and include the constant of integration, denoted as \(C\).
**Graph or Diagram:**
There is a simple rectangular box below the text without any additional information or details provided within it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5ddfa6d-de49-474d-854c-ce697169e6c0%2Fba095592-224c-430d-8d3f-55b869947fff%2Fq44mhqp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To evaluate the indefinite integral
\[
\int \left( 3 + \frac{1}{t} \right)^6 \frac{1}{t^2} dt
\]
Use C for the constant of integration.
**Explanation:**
There is a mathematical expression given in the form of an indefinite integral. The integral involves the function \(\left( 3 + \frac{1}{t} \right)^6\) divided by \(t^2\), and the variable of integration is \(t\). The task is to find the integral and include the constant of integration, denoted as \(C\).
**Graph or Diagram:**
There is a simple rectangular box below the text without any additional information or details provided within it.
![**Problem Statement:**
Use an appropriate \( u \)-substitution to evaluate the indefinite integral
\[
\int \left(3 + \frac{1}{t}\right)^6 \frac{1}{t^2} \, dt.
\]
Use \( C \) for the constant of integration.
**Answer:**
[This section is left blank for students to input their solution or for the website to provide a detailed solution.]
**Explanation and Approach:**
- Begin by identifying a suitable \( u \)-substitution that simplifies the integral.
- Typically, choose \( u \) to be a component inside the function being raised to a power. Here, let \( u = 3 + \frac{1}{t} \).
- Differentiate \( u \) with respect to \( t \) and solve for \( dt \) in terms of \( du \) and \( t \).
This tool assists in teaching students how to perform \( u \)-substitution for complex integrals, enhancing their problem-solving skills by showing a systematic approach.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5ddfa6d-de49-474d-854c-ce697169e6c0%2Fba095592-224c-430d-8d3f-55b869947fff%2F5cww91n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use an appropriate \( u \)-substitution to evaluate the indefinite integral
\[
\int \left(3 + \frac{1}{t}\right)^6 \frac{1}{t^2} \, dt.
\]
Use \( C \) for the constant of integration.
**Answer:**
[This section is left blank for students to input their solution or for the website to provide a detailed solution.]
**Explanation and Approach:**
- Begin by identifying a suitable \( u \)-substitution that simplifies the integral.
- Typically, choose \( u \) to be a component inside the function being raised to a power. Here, let \( u = 3 + \frac{1}{t} \).
- Differentiate \( u \) with respect to \( t \) and solve for \( dt \) in terms of \( du \) and \( t \).
This tool assists in teaching students how to perform \( u \)-substitution for complex integrals, enhancing their problem-solving skills by showing a systematic approach.
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