To Evaluate Integrals Symbolically

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To Evaluate Integrals Symbolically

1. Insert the integral operator.

2. Type the expression in the placeholder to the right of the integral sign.

3. Type the integration variable x in the placeholder to the right of the symbol d.

4. Insert the symbolic evaluation operator.

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  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <mult />
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>2</real>
            </apply>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <placeholder />
        </lowerBound>
        <upperBound>
          <placeholder />
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <mult />
          <parens>
            <apply>
              <plus />
              <apply>
                <minus />
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="VARIABLE">x</id>
                  <real>2</real>
                </apply>
                <apply>
                  <mult />
                  <real>2</real>
                  <id xml:space="preserve" labels="VARIABLE">x</id>
                </apply>
              </apply>
              <real>2</real>
            </apply>
          </parens>
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT">e</id>
            <id xml:space="preserve" labels="VARIABLE">x</id>
          </apply>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

You can only evaluate the indefinite integral using the symbolic evaluation operator.

5. To take the definite integral of the same function, from 0 through 5, type 0 in the bottom placeholder of the integral. Type 5 in the top placeholder and repeat steps 2-4.

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  <math resultRef="11">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <mult />
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>2</real>
            </apply>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <real>5</real>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <minus />
          <apply>
            <mult />
            <real>17</real>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT">e</id>
              <real>5</real>
            </apply>
          </apply>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

6. To take the definite integral of the same function when at least one of the limits is an undefined variable, repeat steps 2-4. Then, insert the lower and upper limits of the integral.

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  <math resultRef="12">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <mult />
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>2</real>
            </apply>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <id xml:space="preserve">t</id>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <minus />
          <apply>
            <mult />
            <parens>
              <apply>
                <plus />
                <apply>
                  <minus />
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                    <real>2</real>
                  </apply>
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                  </apply>
                </apply>
                <real>2</real>
              </apply>
            </parens>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT">e</id>
              <id xml:space="preserve" labels="VARIABLE">t</id>
            </apply>
          </apply>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

When evaluating a definite integral in which at least one of the limits is undefined, the symbolic engine assumes the upper limit is greater than the lower limit.

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  <math resultRef="13">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <mult />
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>2</real>
            </apply>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <id xml:space="preserve">a</id>
        </lowerBound>
        <upperBound>
          <id xml:space="preserve">b</id>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <mult />
            <parens>
              <apply>
                <plus />
                <apply>
                  <minus />
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">b</id>
                    <real>2</real>
                  </apply>
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">b</id>
                  </apply>
                </apply>
                <real>2</real>
              </apply>
            </parens>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT">e</id>
              <id xml:space="preserve" labels="VARIABLE">b</id>
            </apply>
          </apply>
          <apply>
            <mult />
            <parens>
              <apply>
                <minus />
                <apply>
                  <minus />
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">a</id>
                  </apply>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="VARIABLE">a</id>
                    <real>2</real>
                  </apply>
                </apply>
                <real>2</real>
              </apply>
            </parens>
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT">e</id>
              <id xml:space="preserve" labels="VARIABLE">a</id>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

7. To assign results of symbolic evaluation to a function, define a function as follows:

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  <math resultRef="14">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id xml:space="preserve" labels="VARIABLE">f</id>
        <boundVars>
          <id xml:space="preserve" labels="VARIABLE">t</id>
        </boundVars>
      </function>
      <symEval>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id xml:space="preserve">x</id>
            </boundVars>
            <apply>
              <mult />
              <apply>
                <pow />
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                <real>2</real>
              </apply>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <real>0</real>
          </lowerBound>
          <upperBound>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t</id>
          </upperBound>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <minus />
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <minus />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                      <real>2</real>
                    </apply>
                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <id xml:space="preserve" labels="VARIABLE">t</id>
              </apply>
            </apply>
            <real>2</real>
          </apply>
        </symResult>
      </symEval>
    </define>
  </math>
</region>

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id xml:space="preserve" labels="VARIABLE">g</id>
        <boundVars>
          <id xml:space="preserve" labels="VARIABLE">a</id>
          <id xml:space="preserve" labels="VARIABLE">b</id>
        </boundVars>
      </function>
      <symEval>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id xml:space="preserve">x</id>
            </boundVars>
            <apply>
              <mult />
              <apply>
                <pow />
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                <real>2</real>
              </apply>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              </apply>
            </apply>
          </lambda>
          <lowerBound>
            <id xml:space="preserve">a</id>
          </lowerBound>
          <upperBound>
            <id xml:space="preserve">b</id>
          </upperBound>
        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <minus />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">b</id>
                      <real>2</real>
                    </apply>
                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve" labels="VARIABLE">b</id>
                    </apply>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <id xml:space="preserve" labels="VARIABLE">b</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <minus />
                  <apply>
                    <minus />
                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve" labels="VARIABLE">a</id>
                    </apply>
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">a</id>
                      <real>2</real>
                    </apply>
                  </apply>
                  <real>2</real>
                </apply>
              </parens>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <id xml:space="preserve" labels="VARIABLE">a</id>
              </apply>
            </apply>
          </apply>
        </symResult>
      </symEval>
    </define>
  </math>
</region>

Cauchy Principal Value

The Cauchy principal value of the integral about a point c, in the interval (a,b), is defined by the following expression:

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    <apply xmlns="http://schemas.mathsoft.com/math50">
      <limit direction="right" />
      <lambda>
        <boundVars>
          <id xml:space="preserve">ε</id>
        </boundVars>
        <parens>
          <apply>
            <plus />
            <apply>
              <integral />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <apply>
                  <id xml:space="preserve">f</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </lambda>
              <lowerBound>
                <id xml:space="preserve">a</id>
              </lowerBound>
              <upperBound>
                <apply>
                  <minus />
                  <id xml:space="preserve">c</id>
                  <id xml:space="preserve">ε</id>
                </apply>
              </upperBound>
            </apply>
            <apply>
              <integral />
              <lambda>
                <boundVars>
                  <id xml:space="preserve">x</id>
                </boundVars>
                <apply>
                  <id xml:space="preserve">f</id>
                  <id xml:space="preserve">x</id>
                </apply>
              </lambda>
              <lowerBound>
                <apply>
                  <plus />
                  <id xml:space="preserve">c</id>
                  <id xml:space="preserve">ε</id>
                </apply>
              </lowerBound>
              <upperBound>
                <id xml:space="preserve">b</id>
              </upperBound>
            </apply>
          </apply>
        </parens>
      </lambda>
      <real>0</real>
    </apply>
  </math>
</region>

The purpose of the Cauchy principal value is to define the value of the integral when the function f has a singularity point at c. For example, the following integral has a singularity point at c=0, and the symbolic evaluation returns an error.

Click to copy this expression

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  <math resultRef="18">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>3</real>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>-1</real>
        </lowerBound>
        <upperBound>
          <real>1</real>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <id xml:space="preserve" labels="*">undefined</id>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

To find the Cauchy principal value of the integral, add the modifier cauchy.

Click to copy this expression

Symbolic evaluation of integrals that require Cauchy analysis return a result of undefined unless the cauchy keyword is specified.

Related Topics

About Symbolic Calculus

To Evaluate Derivatives Symbolically

Cauchy Distribution

Example: Symbolic Calculus I

Example: Symbolic Calculus II

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