Friday, May 31, 2019

The definite integral and the integration by substitution

Objective:


Apply the method of integration by substitution to the resolution of definite integrals


Method (Recall)


The method by substitution consists in substituting the given variable by another variable
that makes the integral easier to calculate.


Let’s integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x222B;</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></math>by substitution.


If this integral can be written <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x222B;</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></math>with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">u</mi><mo>=</mo><mi mathvariant="normal">g</mi><mfenced><mi mathvariant="normal">x</mi></mfenced></math>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>du</mi><mo>=</mo><mi mathvariant="normal">g</mi><mo>'</mo><mfenced><mi mathvariant="normal">x</mi></mfenced><mi>dx</mi></math> then


<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x222B;</mo><mi>f</mi><mfenced><mrow><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mo>&#x222B;</mo><mi>f</mi><mfenced><mi>u</mi></mfenced><mi>d</mi><mi>u</mi></math>


The integral <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x222B;</mo><mi>f</mi><mfenced><mi>u</mi></mfenced><mi>d</mi><mi>u</mi></math> becomes easier to calculate. In the case of a definite integral
we have to calculate the new limits of integration for u.


Example I


Let’s calculate  <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>1</mn></msubsup><msqrt><mn>3</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>&#xA0;</mo></msqrt><mo>&#xA0;</mo><mn>2</mn><mi>x</mi><mi>d</mi><mi>x</mi><mspace linebreak="newline"/></math>


<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mi>e</mi><mi>t</mi><mo>'</mo><mi>s</mi><mo>&#xA0;</mo><mi>w</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>e</mi><mo>&#xA0;</mo><mi>u</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>&#xA0;</mo><mi>f</mi><mfenced><mi>u</mi></mfenced><mo>=</mo><mo>&#xA0;</mo><msqrt><mn>3</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>&#xA0;</mo><mo>&#xA0;</mo></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>u</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>2</mn><mi>x</mi><mi>d</mi><mi>x</mi><mo>&#xA0;</mo></math>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mi>e</mi><mi>t</mi><mo>'</mo><mi>s</mi><mo>&#xA0;</mo><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi><mi>u</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>e</mi><mo>&#xA0;</mo><mi>t</mi><mi>h</mi><mi>e</mi><mo>&#xA0;</mo><mi>n</mi><mi>e</mi><mi>w</mi><mo>&#xA0;</mo><mi>l</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>t</mi><mi>s</mi><mo>&#xA0;</mo><mi>f</mi><mi>o</mi><mi>r</mi><mo>&#xA0;</mo><mi>u</mi><mo>.</mo><mo>&#xA0;</mo><mi>F</mi><mi>o</mi><mi>r</mi><mo>&#xA0;</mo><mi>x</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn><mo>&#xA0;</mo><mi>w</mi><mi>e</mi><mo>&#xA0;</mo><mi>h</mi><mi>a</mi><mi>v</mi><mi>e</mi><mo>&#xA0;</mo><mi>u</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>3</mn><mo>.</mo><mo>&#xA0;</mo><mi>F</mi><mi>o</mi><mi>r</mi><mo>&#xA0;</mo><mi>x</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>1</mn><mo>&#xA0;</mo><mi>w</mi><mi>e</mi><mo>&#xA0;</mo><mi>h</mi><mi>a</mi><mi>v</mi><mi>e</mi><mo>&#xA0;</mo><mi>u</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>4</mn><mspace linebreak="newline"/></math>
<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>1</mn></msubsup><msqrt><mn>3</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>&#xA0;</mo><mn>2</mn><mi>x</mi><mo>d</mo><mi>x</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msubsup><mo>&#x222B;</mo><mn>3</mn><mn>4</mn></msubsup><mi>f</mi><mfenced><mi>u</mi></mfenced><mo>d</mo><mi>u</mi></math>
                              = <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>3</mn><mn>4</mn></msubsup><msqrt><mi>u</mi></msqrt><mo>d</mo><mi>u</mi></math>
                              = <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>3</mn><mn>4</mn></msubsup><msup><mi>u</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math>
                             = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi mathvariant="normal">U</mi><mrow><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mn>1</mn></mrow></msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mfenced open="|" close=""><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math>
                          = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi mathvariant="normal">U</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mfrac><mfenced open="|" close=""><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math>
                        
                         = <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="|" close="|"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><msup><mi>u</mi><mn>3</mn></msup></msqrt></mrow></mfenced><mn>3</mn><mn>4</mn></msubsup></math>
                        =
                         <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><msup><mn>4</mn><mn>3</mn></msup></msqrt><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><msup><mn>3</mn><mn>3</mn></msup></msqrt><mspace linebreak="newline"/><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><mn>4</mn><mo>&#xD7;</mo><mn>4</mn><mo>&#xD7;</mo><mn>4</mn><mo>&#xA0;</mo></msqrt><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><mn>3</mn><mo>&#xD7;</mo><mn>3</mn><mo>&#xD7;</mo><mn>3</mn></msqrt><mspace linebreak="newline"/><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>&#xD7;</mo><mn>8</mn><mo>&#xA0;</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msqrt><mn>9</mn><mo>&#xD7;</mo><mn>3</mn></msqrt><mspace linebreak="newline"/><mfrac><mn>16</mn><mn>3</mn></mfrac><mo>&#xA0;</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>&#xD7;</mo><mn>3</mn><mo>&#xD7;</mo><msqrt><mn>3</mn></msqrt><mspace linebreak="newline"/><mfrac><mn>16</mn><mn>3</mn></mfrac><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></math>
                        
                       
Example II Evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>0</mn><mn>2</mn></msubsup><msup><mi>x</mi><mn>2</mn></msup><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup><mi>d</mi><mi>x</mi></math>

We can rewrite the expression as:

We have here 2 separate expressions <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup></math> and x²

We seek to find out if the expression closed to dx is the derivative of the expression x³ in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup></math> .

We notice that is almost the derivative of . It misses the term 3. Therefore we have
to multiply and divide by 3 in order to have 3 . The given expression becomes:


<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup></math> represents a function composite of the exponential function f(x) = <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>x</mi></msup></math> and the function g(x) =

We can write  <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup></math> =  f(g(x)). 3x represents the derivative of x³. We can write   3x².= . g'(x)

Then we have:               

If we do a change of variable by writing u = x³ du/dx = 3x² du = 3x²dx.= g'(x)dx

 f(g(x)) = <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><msup><mi>x</mi><mn>3</mn></msup></msup></math>. becomes f(u) = .<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>u</mi></msup></math>. There are new limits of integration for the new function of u

In the expresion u = x³, for x = 0 u = 0, for x = 2 u = (2)³ = 8

Let's substitute f(g(x)) by f(u) and g'(x)dx by du:

Let's substitute f(u) by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>u</mi></msup></math>:

   
                                      
N.B. We have solved the second example by writing the given function as a composite function. We can skip these steps by doing the change of variable right away.as it was done in the first example,  

Practice

Solve:
 
Interested more about learning integrals visit Center for Integral development





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