Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use absolute values where appropriate.) 4 f(x) 5 3 x > 0 F(x) =
Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use absolute values where appropriate.) 4 f(x) 5 3 x > 0 F(x) =
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
![**Problem Statement:**
Find the most general antiderivative of the function. (Check your answer by differentiation. Use \( C \) for the constant of the antiderivative. Remember to use absolute values where appropriate.)
Given function:
\[ f(x) = \frac{4}{5} - \frac{3}{x}, \quad x > 0 \]
**Solution:**
To find the most general antiderivative \( F(x) \), we will integrate the given function:
1. Integrate each term separately:
\[
\int \left(\frac{4}{5}\right) \, dx = \frac{4}{5}x + C_1
\]
\[
\int \left(-\frac{3}{x}\right) \, dx = -3 \ln|x| + C_2
\]
2. Combine into the general antiderivative:
\[
F(x) = \frac{4}{5}x - 3 \ln|x| + C
\]
where \( C \) is the constant of integration, encompassing \( C_1 + C_2 \).
**Note:** Remember to use absolute values in the logarithm since \( x > 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a99fabf-a31f-4250-98b6-bdb0c3640967%2F11d56535-552d-40bc-925d-40f04c7037c4%2Fs1aixlc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the most general antiderivative of the function. (Check your answer by differentiation. Use \( C \) for the constant of the antiderivative. Remember to use absolute values where appropriate.)
Given function:
\[ f(x) = \frac{4}{5} - \frac{3}{x}, \quad x > 0 \]
**Solution:**
To find the most general antiderivative \( F(x) \), we will integrate the given function:
1. Integrate each term separately:
\[
\int \left(\frac{4}{5}\right) \, dx = \frac{4}{5}x + C_1
\]
\[
\int \left(-\frac{3}{x}\right) \, dx = -3 \ln|x| + C_2
\]
2. Combine into the general antiderivative:
\[
F(x) = \frac{4}{5}x - 3 \ln|x| + C
\]
where \( C \) is the constant of integration, encompassing \( C_1 + C_2 \).
**Note:** Remember to use absolute values in the logarithm since \( x > 0 \).
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