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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![On this Educational website, we explore the process of finding the indefinite integral of composite functions. Below is a sample problem requiring this technique:
\[ \int x^9 \arccsc(x^{10} + 13) \, dx \]
Here, we are tasked with integrating the function \( x^9 \arccsc(x^{10} + 13) \) with respect to \( x \).
### Key Concepts:
1. **Power Functions:** The integrand includes a power function \( x^9 \), which highlights that understanding the rules of integrating polynomials is vital.
2. **Inverse Trigonometric Functions:** The function \( \arccsc \), the inverse cosecant, is part of the integrand, which may require knowledge of specific integration techniques involving inverse trigonometric functions.
3. **Composite Functions:** Recognizing how to handle composite functions \( x^{10} + 13 \) within the context of integration will be necessary.
### Possible Approach:
To approach this integral, one might consider substitution methods due to the complexity introduced by the \( x^{10} \) term inside the inverse cosecant function.
1. **Substitution:** Let \( u = x^{10} + 13 \). This substitution simplifies the function inside the inverse cosecant.
2. **Differentiation of Substitution:** Compute \( du \). Since \( u = x^{10} + 13 \), \( \frac{du}{dx} = 10x^9 \). This helps to express \( dx \) in terms of \( du \).
This integral demonstrates the multi-step approach often necessary in calculus, combining derivative knowledge with inverse trigonometric and polynomial integration techniques.
### Detailed Explanation:
1. **Perform the Substitution:**
\[
u = x^{10} + 13 \quad \Rightarrow \quad du = 10x^9 dx
\]
Thus, \( dx = \frac{du}{10x^9} \).
2. **Rewrite the Integral in Terms of \( u \):**
\[
\int x^9 \arccsc(u) \cdot \frac{du}{10x^9}
\]
Simplifies to:
\[
\int \frac{1}{10} \arccsc(u) \, du
\]
3. **Evaluate the New Integral:**
\[
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F84a04708-ec85-4a50-8a9a-f29e8c2061e6%2Fd1ojr46.jpeg&w=3840&q=75)
Transcribed Image Text:On this Educational website, we explore the process of finding the indefinite integral of composite functions. Below is a sample problem requiring this technique:
\[ \int x^9 \arccsc(x^{10} + 13) \, dx \]
Here, we are tasked with integrating the function \( x^9 \arccsc(x^{10} + 13) \) with respect to \( x \).
### Key Concepts:
1. **Power Functions:** The integrand includes a power function \( x^9 \), which highlights that understanding the rules of integrating polynomials is vital.
2. **Inverse Trigonometric Functions:** The function \( \arccsc \), the inverse cosecant, is part of the integrand, which may require knowledge of specific integration techniques involving inverse trigonometric functions.
3. **Composite Functions:** Recognizing how to handle composite functions \( x^{10} + 13 \) within the context of integration will be necessary.
### Possible Approach:
To approach this integral, one might consider substitution methods due to the complexity introduced by the \( x^{10} \) term inside the inverse cosecant function.
1. **Substitution:** Let \( u = x^{10} + 13 \). This substitution simplifies the function inside the inverse cosecant.
2. **Differentiation of Substitution:** Compute \( du \). Since \( u = x^{10} + 13 \), \( \frac{du}{dx} = 10x^9 \). This helps to express \( dx \) in terms of \( du \).
This integral demonstrates the multi-step approach often necessary in calculus, combining derivative knowledge with inverse trigonometric and polynomial integration techniques.
### Detailed Explanation:
1. **Perform the Substitution:**
\[
u = x^{10} + 13 \quad \Rightarrow \quad du = 10x^9 dx
\]
Thus, \( dx = \frac{du}{10x^9} \).
2. **Rewrite the Integral in Terms of \( u \):**
\[
\int x^9 \arccsc(u) \cdot \frac{du}{10x^9}
\]
Simplifies to:
\[
\int \frac{1}{10} \arccsc(u) \, du
\]
3. **Evaluate the New Integral:**
\[
\
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