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

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   it follows that <math>F'(x)=f(x)</math>
 
   it follows that <math>F'(x)=f(x)</math>
  
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   The antidifferentiation process is also called integration and is denoted by <math>\int</math>
+
   The antidifferentiation process is also called integration and is denoted by <math>\int</math> (integral sign).
 +
  <math>\int f(x)dx</math> is the indefinite integral of <math>f(x)</math>
 +
 
 +
  If <math>F'(x)=f(x)</math> for all <math>x</math>, we can write:
 +
  <math>\int f(x)dx=F(x)+C</math> for <math>C</math> is a constant.
 +
 
 +
==Basic Integration Rules==
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<math>1.\int kdx=kx+C</math> for <math>k</math> is a constant.
 +
 
 +
<math>2.\int kf(x)=k\int f(x)dx</math>
 +
 
 +
<math>3.\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx</math>
 +
 
 +
<math>4.\int [f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx</math>
 +
 
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<math>5.\int x^n dx=\frac{x^{n+1}}{n+1}+C</math> for <math>n\ne -1</math>
 +
 
 +
'''Exercises 1''' 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;"
 +
!Solution: &nbsp;
 +
|-
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|<math>\int 5x^{-3}dx=5\int x^{-3}dx=5\frac{x^{-3+1}}{-3+1}+C=\frac{-5}{2}x^{-2}+C</math>
 +
|}
 +
 
 +
'''Exercises 2''' Solve the initial value problems, given:
 +
 
 +
'''5)''' <math>f'(x)=\frac{1}{5}x-2</math> and <math>f(10)=-10</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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|Notice <math>f(x)=\int f'(x)dx=\int (\frac{1}{5}x-2)dx=\frac{1}{5}\frac{x^2}{2}-2x+C=\frac{1}{10}x^2-2x+C</math>
 +
|-
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|So, <math>f(x)=\frac{1}{10}x^2-2x+C</math>
 +
|-
 +
|we are given <math>f(10)=-10</math>, so <math>\frac{1}{10}(10)^2-2(10)+C=10</math>
 +
|-
 +
|Hence, <math>C=20</math>
 +
|-
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|Therefore, <math>f(x)=\frac{1}{10}x^2-2x+20</math>
 +
|}
 +
 
 +
'''6)''' <math>f'(x)=3x^2+4</math> and <math>f(-1)=-6</math>
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
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|-
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|Notice <math>f(x)=\int f'(x)dx=\int (3x^2+4)dx=x^3+4x+C</math>
 +
|-
 +
|So, <math>f(x)=x^3+4x+C</math>
 +
|-
 +
|we are given <math>f(-1)=-6</math>, so <math>(-1)^3+4(-1)+C=-6</math>
 +
|-
 +
|Hence, <math>C=-1</math>
 +
|-
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|Therefore, <math>f(x)=x^3+4x-1</math>
 +
|}
  
  

Latest revision as of 07:33, 12 August 2020

Antiderivatives

 A function  is an antiderivative of a function  when for every 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}
 in the domain of 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 f}
, 
 it follows that 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 F'(x)=f(x)}


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


 If 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 F'(x)=f(x)}
 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 1 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}+C=\frac{-5}{2}x^{-2}+C}

Exercises 2 Solve the initial value problems, given:

5) 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 f'(x)=\frac{1}{5}x-2} and 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 f(10)=-10}

Solution:  
Notice 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 f(x)=\int f'(x)dx=\int (\frac{1}{5}x-2)dx=\frac{1}{5}\frac{x^2}{2}-2x+C=\frac{1}{10}x^2-2x+C}
So, 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 f(x)=\frac{1}{10}x^2-2x+C}
we are given 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 f(10)=-10} , so 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 \frac{1}{10}(10)^2-2(10)+C=10}
Hence, 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=20}
Therefore, 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 f(x)=\frac{1}{10}x^2-2x+20}

6) 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 f'(x)=3x^2+4} and 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 f(-1)=-6}

Solution:  
Notice 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 f(x)=\int f'(x)dx=\int (3x^2+4)dx=x^3+4x+C}
So, 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 f(x)=x^3+4x+C}
we are given 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 f(-1)=-6} , so 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)^3+4(-1)+C=-6}
Hence, 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=-1}
Therefore, 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 f(x)=x^3+4x-1}


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