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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![### Integral Evaluation Exercises
Below are some integral problems that need evaluation. Each exercise provides an integral expression along with a space to write the solution. These problems are useful for practicing integration techniques.
---
#### Exercise 1
Evaluate:
\[
\int 2x^3 \sqrt{81 + x^2} \, dx
\]
Result:
\[
\int 2x^3 \sqrt{81 + x^2} \, dx = \underline{\hspace{4cm}}
\]
---
#### Exercise 2
Evaluate:
\[
\int_7^{(5 + r^8)^7} \, dr
\]
Result:
\[
\int_7^{(5 + r^8)^7} \, dr = \underline{\hspace{4cm}}
\]
---
#### Exercise 3
Evaluate:
\[
\int_3^{(7 + t^4)^2} \, dt
\]
Result:
\[
\int_3^{(7 + t^4)^2} \, dt = \underline{\hspace{4cm}}
\]
---
#### Exercise 4
Evaluate:
\[
\int_7^{(4 + t^8)^8} \, dt
\]
Result:
\[
\int_7^{(4 + t^8)^8} \, dt = \underline{\hspace{4cm}}
\]
---
#### Exercise 5
Evaluate:
\[
\int x^3 e^{x^4} \, dx
\]
Result:
\[
\int x^3 e^{x^4} \, dx = \underline{\hspace{4cm}}
\]
---
These exercises are designed to strengthen understanding of evaluating definite and indefinite integrals through practice.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb85c101b-d3f5-410c-a5d1-ffd9e9de2493%2F1397453f-fad2-4689-8aca-ad53b7df1637%2F61jmr64_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Evaluation Exercises
Below are some integral problems that need evaluation. Each exercise provides an integral expression along with a space to write the solution. These problems are useful for practicing integration techniques.
---
#### Exercise 1
Evaluate:
\[
\int 2x^3 \sqrt{81 + x^2} \, dx
\]
Result:
\[
\int 2x^3 \sqrt{81 + x^2} \, dx = \underline{\hspace{4cm}}
\]
---
#### Exercise 2
Evaluate:
\[
\int_7^{(5 + r^8)^7} \, dr
\]
Result:
\[
\int_7^{(5 + r^8)^7} \, dr = \underline{\hspace{4cm}}
\]
---
#### Exercise 3
Evaluate:
\[
\int_3^{(7 + t^4)^2} \, dt
\]
Result:
\[
\int_3^{(7 + t^4)^2} \, dt = \underline{\hspace{4cm}}
\]
---
#### Exercise 4
Evaluate:
\[
\int_7^{(4 + t^8)^8} \, dt
\]
Result:
\[
\int_7^{(4 + t^8)^8} \, dt = \underline{\hspace{4cm}}
\]
---
#### Exercise 5
Evaluate:
\[
\int x^3 e^{x^4} \, dx
\]
Result:
\[
\int x^3 e^{x^4} \, dx = \underline{\hspace{4cm}}
\]
---
These exercises are designed to strengthen understanding of evaluating definite and indefinite integrals through practice.
![The image shows a mathematical problem involving an integral. The expression included is:
\[
\int 2x \sqrt[3]{81 + x^2} \, dx
\]
Additionally, there is a second line containing the same integral with an equal sign, indicating the need to solve the expression:
\[
\int 2x \sqrt[3]{81 + x^2} \, dx = \underline{\quad}
\]
The problem is to find the indefinite integral of the given function with respect to \(x\). There are no graphs or diagrams accompanying this expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb85c101b-d3f5-410c-a5d1-ffd9e9de2493%2F1397453f-fad2-4689-8aca-ad53b7df1637%2F90kqdnp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical problem involving an integral. The expression included is:
\[
\int 2x \sqrt[3]{81 + x^2} \, dx
\]
Additionally, there is a second line containing the same integral with an equal sign, indicating the need to solve the expression:
\[
\int 2x \sqrt[3]{81 + x^2} \, dx = \underline{\quad}
\]
The problem is to find the indefinite integral of the given function with respect to \(x\). There are no graphs or diagrams accompanying this expression.
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