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Find the general antiderivative of the function. (If nec- essary, rewrite the function before antidifferentiation.) m(x) 3 + 2x/x %3D V

Find the general antiderivative of the function. (If nec- essary, rewrite the function before antidifferentiation.) m(x) 3 + 2x/x %3D V

Calculus: Early Transcendentals
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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**Problem Statement:** Find the general antiderivative of the function. (If necessary, rewrite the function before antidifferentiation.) \( m(x) = \frac{3}{\sqrt{x}} + 2x\sqrt{x} \) **Instructions:** - Simplify the function if needed for easier antiderivative calculation. - Recall the basic integration rules and apply them to each term separately. - Remember to include the constant of integration in your final answer. **Function Simplification:** 1. Rewrite \( \frac{3}{\sqrt{x}} \) as \( 3x^{-1/2} \). 2. Rewrite \( 2x\sqrt{x} \) as \( 2x^{1/2 + 1} = 2x^{3/2} \). **Calculate Antiderivative:** - Integrate each term: - \( \int 3x^{-1/2} \, dx = 3 \cdot \frac{x^{1/2}}{1/2} = 6x^{1/2} \) - \( \int 2x^{3/2} \, dx = 2 \cdot \frac{x^{5/2}}{5/2} = \frac{4}{5}x^{5/2} \) **General Antiderivative:** Combine the results: \( M(x) = 6x^{1/2} + \frac{4}{5}x^{5/2} + C \) where \( C \) is the constant of integration.
Transcribed Image Text:**Problem Statement:** Find the general antiderivative of the function. (If necessary, rewrite the function before antidifferentiation.) \( m(x) = \frac{3}{\sqrt{x}} + 2x\sqrt{x} \) **Instructions:** - Simplify the function if needed for easier antiderivative calculation. - Recall the basic integration rules and apply them to each term separately. - Remember to include the constant of integration in your final answer. **Function Simplification:** 1. Rewrite \( \frac{3}{\sqrt{x}} \) as \( 3x^{-1/2} \). 2. Rewrite \( 2x\sqrt{x} \) as \( 2x^{1/2 + 1} = 2x^{3/2} \). **Calculate Antiderivative:** - Integrate each term: - \( \int 3x^{-1/2} \, dx = 3 \cdot \frac{x^{1/2}}{1/2} = 6x^{1/2} \) - \( \int 2x^{3/2} \, dx = 2 \cdot \frac{x^{5/2}}{5/2} = \frac{4}{5}x^{5/2} \) **General Antiderivative:** Combine the results: \( M(x) = 6x^{1/2} + \frac{4}{5}x^{5/2} + C \) where \( C \) is the constant of integration.
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