以符號方式計算積分

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以符號方式計算積分

1. 插入積分運算子。

2. 在積分號右側的佔位符號中鍵入運算式。

3. 在符號 d 右側的佔位符號中鍵入積分變數 x。

4. 插入符號計算運算子。

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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>

您只能使用符號計算運算子計算不定積分。

5. 若要求解相同函數的定積分 (0 到 5)，請在積分的底部佔位符號中鍵入 0。在頂部佔位符號中輸入 5，然後重複步驟 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. 若要在至少有一個限制為未定義的變數時採用相同函數的定積分，請重複步驟 2-4。然後，插入積分的上下限。

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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>

當計算之定積分中含至少一個未定義的時，符號引擎會假設上限大於下限。

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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. 若要將符號計算結果指派給函數，請按照以下方式定義函數：

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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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  <math resultRef="15">
    <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>

科西主值

區間 (a,b) 中點 c 的相關積分科西主值，由下列運算式定義：

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  <math resultRef="17">
    <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>

科西主值是為了在函數 f 於 c 處有奇點時，定義積分值之用。例如，下列積分在 c=0 處有奇點，因此符號計算傳回錯誤。

按一下以複製此運算式

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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>

若要求解積分的科西主值，請新增修飾符 cauchy。

按一下以複製此運算式

除非指定了 cauchy 關鍵字，否則需要科西分析之積分的符號計算會傳回 undefined 的結果。

相關主題

關於符號微積分

以符號方式計算導數

科西分佈

範例：符號微積分 I

範例：符號微積分 II

這是否有幫助?