Find the indefinite integral. [(sec e tan ()i + (tan t)3 + (2 sin t cos t)k] dt Integrate the given integral with respect to t on a component-by-component basis. n/6 [" (sec t tan t)i + (tan t)j + (2 sin t cos t)k dt (sec e tan t) dt )i +/" (tan t) dt cos t) dt )k sin t cos - [ n/6, i+ In R/6
Find the indefinite integral. [(sec e tan ()i + (tan t)3 + (2 sin t cos t)k] dt Integrate the given integral with respect to t on a component-by-component basis. n/6 [" (sec t tan t)i + (tan t)j + (2 sin t cos t)k dt (sec e tan t) dt )i +/" (tan t) dt cos t) dt )k sin t cos - [ n/6, i+ In R/6
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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![### Vector Integral Calculation
#### Problem Statement
Find the indefinite integral:
\[
\int_0^{\pi/6} \left((\sec t \tan t) \mathbf{i} + (\tan t) \mathbf{j} + (2 \sin t \cos t) \mathbf{k}\right) \, dt
\]
#### Step-by-step Solution
Integrate the given integral with respect to \( t \) on a component-by-component basis:
\[
\int_0^{\pi/6} \left((\sec t \tan t) \mathbf{i} + (\tan t) \mathbf{j} + (2 \sin t \cos t) \mathbf{k}\right) \, dt
\]
This can be split into:
\[
= \left( \int_0^{\pi/6} (\sec t \tan t) \, dt \right) \mathbf{i} + \left( \int_0^{\pi/6} (\tan t) \, dt \right) \mathbf{j} + \left( \int_0^{\pi/6} (2 \sin t \cos t) \, dt \right) \mathbf{k}
\]
\[
= \left[ \, \right]_0^{\pi/6} \mathbf{i} + \left[ \ln \, \right]_0^{\pi/6} \mathbf{j} + \left[ \, \right]_0^{\pi/6} \mathbf{k}
\]
Here, each component is integrated separately over the interval \([0, \pi/6]\).
### Explanation of Components
1. **First Component (\(\mathbf{i}\)):**
- Integral of \(\sec t \tan t\) from \(0\) to \(\pi/6\).
2. **Second Component (\(\mathbf{j}\)):**
- Integral of \(\tan t\) from \(0\) to \(\pi/6\).
- Results in a logarithmic expression \(\ln\).
3. **Third Component (\(\mathbf{k}\)):**
- Integral of \(2 \sin t \cos t\) from \(0\) to \(\pi/6\).
- This simplifies through trigonometric identities.
The solution involves performing the definite integrations and substituting within the brackets](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc08bab33-dfc3-4475-87c9-eef4dfb69a19%2F9f997ca8-a7b3-413c-b6a4-232485acfdb0%2F6tsqgm_processed.png&w=3840&q=75)
Transcribed Image Text:### Vector Integral Calculation
#### Problem Statement
Find the indefinite integral:
\[
\int_0^{\pi/6} \left((\sec t \tan t) \mathbf{i} + (\tan t) \mathbf{j} + (2 \sin t \cos t) \mathbf{k}\right) \, dt
\]
#### Step-by-step Solution
Integrate the given integral with respect to \( t \) on a component-by-component basis:
\[
\int_0^{\pi/6} \left((\sec t \tan t) \mathbf{i} + (\tan t) \mathbf{j} + (2 \sin t \cos t) \mathbf{k}\right) \, dt
\]
This can be split into:
\[
= \left( \int_0^{\pi/6} (\sec t \tan t) \, dt \right) \mathbf{i} + \left( \int_0^{\pi/6} (\tan t) \, dt \right) \mathbf{j} + \left( \int_0^{\pi/6} (2 \sin t \cos t) \, dt \right) \mathbf{k}
\]
\[
= \left[ \, \right]_0^{\pi/6} \mathbf{i} + \left[ \ln \, \right]_0^{\pi/6} \mathbf{j} + \left[ \, \right]_0^{\pi/6} \mathbf{k}
\]
Here, each component is integrated separately over the interval \([0, \pi/6]\).
### Explanation of Components
1. **First Component (\(\mathbf{i}\)):**
- Integral of \(\sec t \tan t\) from \(0\) to \(\pi/6\).
2. **Second Component (\(\mathbf{j}\)):**
- Integral of \(\tan t\) from \(0\) to \(\pi/6\).
- Results in a logarithmic expression \(\ln\).
3. **Third Component (\(\mathbf{k}\)):**
- Integral of \(2 \sin t \cos t\) from \(0\) to \(\pi/6\).
- This simplifies through trigonometric identities.
The solution involves performing the definite integrations and substituting within the brackets
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