Difference between revisions of "Math 22 Antiderivatives and Indefinite Integrals"

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<math>5.\int x^n dx=\frac{x^{n+1}}{n+1}+C</math> for <math>n\ne -1</math>
 
<math>5.\int x^n dx=\frac{x^{n+1}}{n+1}+C</math> for <math>n\ne -1</math>
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'''Exercises''' Find the indefinite integral
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'''1)''' <math>\int 7dr</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\int 7dr=7r+C</math>
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|}
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'''2)''' <math>\int -4dx</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\int -4dx=-4x+C</math>
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|}
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'''3)''' <math>\int 7x^2dx</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\int 7x^2dx=7\int x^2dx=7\frac{x^{2+1}}{2+1}+C=\frac{7}{3}x^3+C</math>
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|}
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'''4)''' <math>\int 5x^{-3}dx</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\int 5x^{-3}dx=5\int x^{-3}dx=5\frac{x^{-3+1}}{-3+1}=\frac{-5}{2}x^{-2}</math>
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|}
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Revision as of 08:23, 12 August 2020

Antiderivatives

 A function  is an antiderivative of a function  when for every  in the domain of , 
 it follows that 

 The antidifferentiation process is also called integration and is denoted by  (integral sign).
  is the indefinite integral of 

 If  for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
, we can write:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int f(x)dx=F(x)+C}
 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}
 is a constant.

Basic Integration Rules

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1.\int kdx=kx+C} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is a constant.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2.\int kf(x)=k\int f(x)dx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3.\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4.\int [f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5.\int x^n dx=\frac{x^{n+1}}{n+1}+C} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ne -1}

Exercises Find the indefinite integral

1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 7dr}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 7dr=7r+C}

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int -4dx}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int -4dx=-4x+C}

3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 7x^2dx}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 7x^2dx=7\int x^2dx=7\frac{x^{2+1}}{2+1}+C=\frac{7}{3}x^3+C}

4) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 5x^{-3}dx}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 5x^{-3}dx=5\int x^{-3}dx=5\frac{x^{-3+1}}{-3+1}=\frac{-5}{2}x^{-2}}

Return to Topics Page

This page were made by Tri Phan

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