Consider the following. de Use the substitution u = sin(0) to rewrite the given integral. SCC du The resulting integral can be evaluated using which of the following entries from the table of integ entry 54 entry 55 entry 56 entry 57 entry 58 Determine the values of a and b. a = = Evaluate the integral and state the result in terms of 0. (Use C for the constant of integration.) 5 sin(20) √4-5 sin(0)
Consider the following. de Use the substitution u = sin(0) to rewrite the given integral. SCC du The resulting integral can be evaluated using which of the following entries from the table of integ entry 54 entry 55 entry 56 entry 57 entry 58 Determine the values of a and b. a = = Evaluate the integral and state the result in terms of 0. (Use C for the constant of integration.) 5 sin(20) √4-5 sin(0)
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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![### Integration Examples of Functions Involving Radicals
Here are some integrals involving functions with radicals and the detailed solutions.
#### Example 54
\[
\int u \sqrt{a + bu} \, du = \frac{2}{15b^2} (3bu - 2a)(a + bu)^{3/2} + C
\]
#### Example 55
\[
\int \frac{u \, du}{\sqrt{a + bu}} = \frac{2}{3b^2} (bu - 2a) \sqrt{a + bu} + C
\]
#### Example 56
\[
\int \frac{u^2 \, du}{\sqrt{a + bu}} = \frac{2}{15b^3} \left( 8a^2 + 3b^2u^2 - 4abu \right) \sqrt{a + bu} + C
\]
#### Example 57
\[
\int \frac{du}{u \sqrt{a + bu}} = \begin{cases}
\frac{1}{\sqrt{a}} \ln \left| \frac{\sqrt{a + bu} - \sqrt{a}}{\sqrt{a + bu} + \sqrt{a}} \right| + C, & \text{if } a > 0 \\
\frac{2}{\sqrt{-a}} \tan^{-1} \left( \sqrt{\frac{a + bu}{-a}} \right) + C, & \text{if } a < 0
\end{cases}
\]
#### Example 58
\[
\int \frac{\sqrt{a + bu}}{u} \, du = 2 \sqrt{a + bu} + a \int \frac{du}{u \sqrt{a + bu}}
\]
### Notes:
- Each integral involves functions containing radicals.
- Different techniques, such as substitution, are utilized to simplify and solve the integrals.
- Example 57 provides two different forms of the solution based on the sign of \(a\).
- Example 58 indicates a combination approach where part of the integral \( \int \frac{du}{u \sqrt{a + bu}} \) is required to complete the integration process.
This collection of integrals serves as a reference for managing](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2F052c9ed2-81db-40bc-ab13-22a8ae631fe3%2Fk3p3l1k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integration Examples of Functions Involving Radicals
Here are some integrals involving functions with radicals and the detailed solutions.
#### Example 54
\[
\int u \sqrt{a + bu} \, du = \frac{2}{15b^2} (3bu - 2a)(a + bu)^{3/2} + C
\]
#### Example 55
\[
\int \frac{u \, du}{\sqrt{a + bu}} = \frac{2}{3b^2} (bu - 2a) \sqrt{a + bu} + C
\]
#### Example 56
\[
\int \frac{u^2 \, du}{\sqrt{a + bu}} = \frac{2}{15b^3} \left( 8a^2 + 3b^2u^2 - 4abu \right) \sqrt{a + bu} + C
\]
#### Example 57
\[
\int \frac{du}{u \sqrt{a + bu}} = \begin{cases}
\frac{1}{\sqrt{a}} \ln \left| \frac{\sqrt{a + bu} - \sqrt{a}}{\sqrt{a + bu} + \sqrt{a}} \right| + C, & \text{if } a > 0 \\
\frac{2}{\sqrt{-a}} \tan^{-1} \left( \sqrt{\frac{a + bu}{-a}} \right) + C, & \text{if } a < 0
\end{cases}
\]
#### Example 58
\[
\int \frac{\sqrt{a + bu}}{u} \, du = 2 \sqrt{a + bu} + a \int \frac{du}{u \sqrt{a + bu}}
\]
### Notes:
- Each integral involves functions containing radicals.
- Different techniques, such as substitution, are utilized to simplify and solve the integrals.
- Example 57 provides two different forms of the solution based on the sign of \(a\).
- Example 58 indicates a combination approach where part of the integral \( \int \frac{du}{u \sqrt{a + bu}} \) is required to complete the integration process.
This collection of integrals serves as a reference for managing
![**Consider the following.**
\[ \int \frac{\sin(2\theta)}{\sqrt{4 - 5 \sin(\theta)}} \, d\theta \]
**Use the substitution \( u = \sin(\theta) \) to rewrite the given integral.**
\[ \int \left( \text{(expression involving } u \text{)} \right) \, du \]
**The resulting integral can be evaluated using which of the following entries from the table of integrals?**
- [ ] entry 54
- [ ] entry 55
- [ ] entry 56
- [ ] entry 57
- [ ] entry 58
**Determine the values of \( a \) and \( b \).**
\[ a = \underline{\hspace{2cm}} \]
\[ b = \underline{\hspace{2cm}} \]
**Evaluate the integral and state the result in terms of \( \theta \). (Use \( C \) for the constant of integration.)**
\[ \underline{\hspace{12cm}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2F052c9ed2-81db-40bc-ab13-22a8ae631fe3%2Fa7m3j7l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the following.**
\[ \int \frac{\sin(2\theta)}{\sqrt{4 - 5 \sin(\theta)}} \, d\theta \]
**Use the substitution \( u = \sin(\theta) \) to rewrite the given integral.**
\[ \int \left( \text{(expression involving } u \text{)} \right) \, du \]
**The resulting integral can be evaluated using which of the following entries from the table of integrals?**
- [ ] entry 54
- [ ] entry 55
- [ ] entry 56
- [ ] entry 57
- [ ] entry 58
**Determine the values of \( a \) and \( b \).**
\[ a = \underline{\hspace{2cm}} \]
\[ b = \underline{\hspace{2cm}} \]
**Evaluate the integral and state the result in terms of \( \theta \). (Use \( C \) for the constant of integration.)**
\[ \underline{\hspace{12cm}} \]
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