Difference between revisions of "Math 22 Integration by Parts and Present Value"

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   <math>\int u dv=uv-\int v du</math>
 
   <math>\int u dv=uv-\int v du</math>
  
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'''Exercises''' Use integration by parts to evaluation:
  
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'''1)''' <math>\int \ln x 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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|Let <math>u=\ln x</math>, <math>>du=\frac{1}{x}dx</math>
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|-
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|and <math>dv=dx</math> and <math>v=x</math>
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|-
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|Then, by integration by parts: <math>\int \ln x dx=x\ln x-\int x\frac{1}{x}dx=x\ln x-\int dx=x\ln x -x +C</math>
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|}
  
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'''2)''' <math>\int xe^{3x}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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|Let <math>u=x</math>, <math>du=dx</math>
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|-
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|and <math>dv=e^{3x}dx</math> and <math>v=\frac{1}{3}e^{3x}</math>
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|-
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|Then, by integration by parts: <math>\int xe^{3x}dx=x\frac{1}{3}e^{3x} -\int\frac{1}{3}e^{3x} dx=x\frac{1}{3}e^{3x}-\frac{1}{9}e^{3x} +C </math>
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|}
  
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'''3)''' <math>\int x^2e^{-x}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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|Let <math>u=x^2</math>, <math>du=2xdx</math>
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|-
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|and <math>dv=e^{-x}dx</math> and <math>v=-e^{-x}</math>
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|-
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|Then, by integration by parts: <math>\int x^2e^{-x}dx=x^2(-e^{-x}) -\int-e^{-x}2x dx=-x^2e^{-x}+\int 2xe^{-x}dx </math>
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|-
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|Now, we apply integration by parts the second time for <math>\int 2xe^{-x}dx</math>
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|-
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|Let <math>u=2x</math>, <math>du=2dx</math>
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|-
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|and <math>dv=e^{-x}dx</math> and <math>v=-e^{-x}</math>
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|-
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|So <math>\int 2xe^{-x}dx=2x(-e^{-x})-\int -e^{-x} 2dx=-2xe^{-x}-e^{-x}+C</math>
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|-
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|Therefore, <math>\int x^2e^{-x}dx=-x^2e^{-x}-2xe^{-x}-e^{-x}+C</math>
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|}
  
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==Note==
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1. Tabular integration technique (look it up) is convenient in some cases.
  
  

Latest revision as of 17:22, 3 September 2020

Integration by Parts

 Let 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 u}
 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 v}
 be differentiable functions 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 x}
.
 
 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 u dv=uv-\int v du}

Exercises Use integration by parts to evaluation:

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 \ln x dx}

Solution:  
Let 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 u=\ln x} , 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 >du=\frac{1}{x}dx}
and 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 v=x}
Then, by integration by parts: 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 \ln x dx=x\ln x-\int x\frac{1}{x}dx=x\ln x-\int dx=x\ln x -x +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 xe^{3x}dx}

Solution:  
Let 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 u=x} , 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 du=dx}
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 dv=e^{3x}dx} 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 v=\frac{1}{3}e^{3x}}
Then, by integration by parts: 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 xe^{3x}dx=x\frac{1}{3}e^{3x} -\int\frac{1}{3}e^{3x} dx=x\frac{1}{3}e^{3x}-\frac{1}{9}e^{3x} +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 x^2e^{-x}dx}

Solution:  
Let 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 u=x^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 du=2xdx}
and 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 v=-e^{-x}}
Then, by integration by parts: 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 x^2e^{-x}dx=x^2(-e^{-x}) -\int-e^{-x}2x dx=-x^2e^{-x}+\int 2xe^{-x}dx }
Now, we apply integration by parts the second time 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 \int 2xe^{-x}dx}
Let 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 u=2x} , 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 du=2dx}
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 dv=e^{-x}dx} 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 v=-e^{-x}}
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 \int 2xe^{-x}dx=2x(-e^{-x})-\int -e^{-x} 2dx=-2xe^{-x}-e^{-x}+C}
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 \int x^2e^{-x}dx=-x^2e^{-x}-2xe^{-x}-e^{-x}+C}

Note

1. Tabular integration technique (look it up) is convenient in some cases.


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