要以符号形式计算积分

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要以符号形式计算积分

1. 插入积分运算符。

2. 在积分符号右侧的占位符中键入表达式。

3. 在符号 d 右侧的占位符中键入积分变量 x。

4. 插入符号求值运算符。

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  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
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          <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>

只能使用符号求值运算符计算不定积分。

5. 要从 0 到 5 计算同一函数的定积分，可在积分的底部占位符中键入 0。在顶部占位符中键入 5，并重复步骤 2-4。

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    <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. 当至少有一个极限值为未定义的变量时，要计算同一函数的定积分，请重复步骤 2-4。然后，插入积分的下限和上限。

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          <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>

在计算至少有一个未定义极限值的定积分时，符号引擎将假定上限大于下限。

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    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
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          <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>
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7. 要将符号运算结果分配给函数，请按如下方式定义函数：

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                <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 />
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              <mult />
              <parens>
                <apply>
                  <plus />
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                    <minus />
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                      <id xml:space="preserve" labels="VARIABLE">t</id>
                      <real>2</real>
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                    <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">
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      </function>
      <symEval>
        <apply>
          <integral />
          <lambda>
            <boundVars>
              <id xml:space="preserve">x</id>
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            <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>
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        </apply>
        <command>
          <placeholder />
        </command>
        <symResult>
          <apply>
            <plus />
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <minus />
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                      <pow />
                      <id xml:space="preserve" labels="VARIABLE">b</id>
                      <real>2</real>
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                    <apply>
                      <mult />
                      <real>2</real>
                      <id xml:space="preserve" labels="VARIABLE">b</id>
                    </apply>
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                  <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>

柯西主值

间隔 (a,b) 中关于点 c 的积分的柯西主值由以下表达式定义：

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    <apply xmlns="http://schemas.mathsoft.com/math50">
      <limit direction="right" />
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          <apply>
            <plus />
            <apply>
              <integral />
              <lambda>
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                  <id xml:space="preserve">x</id>
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                <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>

柯西主值的目的是：当函数 f 在 c 处具有奇点时，定义积分的值。例如，以下积分在 c=0 处具有奇点，并且符号运算会返回错误。

单击可复制此表达式

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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>

要查找积分的柯西主值，可添加修饰符 cauchy。

单击可复制此表达式

除非指定了 cauchy 关键字，需要柯西分析的积分的符号运算会返回 undefined 的结果。

相关主题

关于符号微积分

要以符号形式计算导数

柯西分布

示例：符号微积分 I

示例：符号微积分 II

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