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="6">
    <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.

<region id="ID0EDSHLB" actualWidth="168.5266666666667" actualHeight="58.952000000000005" top="96.000000000000014" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="7">
    <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>

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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  <math resultRef="9">
    <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

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

Click to copy this expression

<region id="ID0EMVHLB" actualWidth="137.65333333333334" actualHeight="80.552" top="403.2000000000001" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="11">
    <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>
        <id xml:space="preserve">cauchy</id>
      </command>
      <symResult>
        <real>0</real>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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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