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
Use patterns to Evaluate the following.
![**Observe the patterns in the following:**
\[
F_0(x) = \frac{1}{1} x (\ln x) - \frac{1}{1^2} x \quad \text{is an antiderivative of} \quad f_0(x) = \left( \ln x \right)
\]
\[
F_1(x) = \frac{1}{2} x^2 (\ln x) - \frac{1}{2^2} x^2 \quad \text{is an antiderivative of} \quad f_1(x) = x \left( \ln x \right)
\]
\[
F_2(x) = \frac{1}{3} x^3 (\ln x) - \frac{1}{3^2} x^3 \quad \text{is an antiderivative of} \quad f_2(x) = x^2 \left( \ln x \right)
\]
\[
F_3(x) = \frac{1}{4} x^4 (\ln x) - \frac{1}{4^2} x^4 \quad \text{is an antiderivative of} \quad f_3(x) = x^3 \left( \ln x \right)
\]
...
In each case above, \( F_n(x) \) represents an antiderivative of \( f_n(x) \), where \( f_n(x) \) is of the form \( x^n (\ln x) \). Notice the pattern in the coefficients and the powers of \( x \). The general form \( F_n(x) \) appears to be constructed by an initial fraction term involving \( x^{n+1} (\ln x) \) minus a fraction term involving \( x^{n+1} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2F0972b8bc-2d9d-448d-b34b-64f8a6984720%2F97fs35q.jpeg&w=3840&q=75)
Transcribed Image Text:**Observe the patterns in the following:**
\[
F_0(x) = \frac{1}{1} x (\ln x) - \frac{1}{1^2} x \quad \text{is an antiderivative of} \quad f_0(x) = \left( \ln x \right)
\]
\[
F_1(x) = \frac{1}{2} x^2 (\ln x) - \frac{1}{2^2} x^2 \quad \text{is an antiderivative of} \quad f_1(x) = x \left( \ln x \right)
\]
\[
F_2(x) = \frac{1}{3} x^3 (\ln x) - \frac{1}{3^2} x^3 \quad \text{is an antiderivative of} \quad f_2(x) = x^2 \left( \ln x \right)
\]
\[
F_3(x) = \frac{1}{4} x^4 (\ln x) - \frac{1}{4^2} x^4 \quad \text{is an antiderivative of} \quad f_3(x) = x^3 \left( \ln x \right)
\]
...
In each case above, \( F_n(x) \) represents an antiderivative of \( f_n(x) \), where \( f_n(x) \) is of the form \( x^n (\ln x) \). Notice the pattern in the coefficients and the powers of \( x \). The general form \( F_n(x) \) appears to be constructed by an initial fraction term involving \( x^{n+1} (\ln x) \) minus a fraction term involving \( x^{n+1} \).
![### Integrals involving \((1 - x)^n (\ln x)\)
Below are several integral expressions, which involve integrating a product of a polynomial term \((1 - x)^n\) and a logarithmic term \(\ln x\), over the interval from \(x = 0\) to \(x = 1\):
1(a)
\[ \int_{x=0}^{1} 2 (1 - x) (\ln x) \, dx = \_\_\_\_\_. \]
1(b)
\[ \int_{x=0}^{1} 3 (1 - x)^2 (\ln x) \, dx = \_\_\_\_\_. \]
1(c)
\[ \int_{x=0}^{1} 4 (1 - x)^3 (\ln x) \, dx = \_\_\_\_\_. \]
1(d)
\[ \int_{x=0}^{1} 5 (1 - x)^4 (\ln x) \, dx = \_\_\_\_\_. \]
1(e)
\[ \int_{x=0}^{1} 6 (1 - x)^5 (\ln x) \, dx = \_\_\_\_\_. \]
\[ 6 \]
These integrals express how one can evaluate integrals of functions that are products of a polynomial and a logarithmic function over a specific interval. Each integral follows the general structure:
\[ \int_{x=0}^{1} n (1 - x)^{(n-1)} (\ln x) \, dx \]
where the coefficient \(n\) and the exponent \((n-1)\) change according to the specific integral being evaluated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2F0972b8bc-2d9d-448d-b34b-64f8a6984720%2F5yntlf3_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Integrals involving \((1 - x)^n (\ln x)\)
Below are several integral expressions, which involve integrating a product of a polynomial term \((1 - x)^n\) and a logarithmic term \(\ln x\), over the interval from \(x = 0\) to \(x = 1\):
1(a)
\[ \int_{x=0}^{1} 2 (1 - x) (\ln x) \, dx = \_\_\_\_\_. \]
1(b)
\[ \int_{x=0}^{1} 3 (1 - x)^2 (\ln x) \, dx = \_\_\_\_\_. \]
1(c)
\[ \int_{x=0}^{1} 4 (1 - x)^3 (\ln x) \, dx = \_\_\_\_\_. \]
1(d)
\[ \int_{x=0}^{1} 5 (1 - x)^4 (\ln x) \, dx = \_\_\_\_\_. \]
1(e)
\[ \int_{x=0}^{1} 6 (1 - x)^5 (\ln x) \, dx = \_\_\_\_\_. \]
\[ 6 \]
These integrals express how one can evaluate integrals of functions that are products of a polynomial and a logarithmic function over a specific interval. Each integral follows the general structure:
\[ \int_{x=0}^{1} n (1 - x)^{(n-1)} (\ln x) \, dx \]
where the coefficient \(n\) and the exponent \((n-1)\) change according to the specific integral being evaluated.
Expert Solution
Step 1
Given integral patterns
Step 2
Use the pattern to integrate each
Step 3
Now use the same integral pattern for the next problem
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