以符號方式計算積分

符號 > 使用符號 > 微積分 > 以符號方式計算積分

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。

按一下以複製此運算式

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  <math resultRef="11" xml:lang="zh_TW">
    <symEval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <mult xml:lang="zh_TW" />
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
              <real xml:lang="zh_TW">2</real>
            </apply>
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">e</id>
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <placeholder xml:lang="zh_TW" />
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <placeholder xml:lang="zh_TW" />
        </upperBound>
      </apply>
      <command xml:lang="zh_TW">
        <placeholder xml:lang="zh_TW" />
      </command>
      <symResult xml:lang="zh_TW">
        <apply xml:lang="zh_TW">
          <mult xml:lang="zh_TW" />
          <apply xml:lang="zh_TW">
            <pow xml:lang="zh_TW" />
            <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">e</id>
            <id labels="*" xml:space="preserve" xml:lang="zh_TW">x</id>
          </apply>
          <parens xml:lang="zh_TW">
            <apply xml:lang="zh_TW">
              <plus xml:lang="zh_TW" />
              <apply xml:lang="zh_TW">
                <minus xml:lang="zh_TW" />
                <apply xml:lang="zh_TW">
                  <pow xml:lang="zh_TW" />
                  <id labels="*" xml:space="preserve" xml:lang="zh_TW">x</id>
                  <real xml:lang="zh_TW">2</real>
                </apply>
                <apply xml:lang="zh_TW">
                  <mult xml:lang="zh_TW" />
                  <real xml:lang="zh_TW">2</real>
                  <id labels="*" xml:space="preserve" xml:lang="zh_TW">x</id>
                </apply>
              </apply>
              <real xml:lang="zh_TW">2</real>
            </apply>
          </parens>
        </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。在頂端佔位符號中鍵入 5，然後重複步驟 2-4。

按一下以複製此運算式

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  <math resultRef="13" xml:lang="zh_TW">
    <symEval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <mult xml:lang="zh_TW" />
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
              <real xml:lang="zh_TW">2</real>
            </apply>
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW" label-is-contextual="true">e</id>
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">0</real>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">5</real>
        </upperBound>
      </apply>
      <command xml:lang="zh_TW">
        <placeholder xml:lang="zh_TW" />
      </command>
      <symResult xml:lang="zh_TW">
        <apply xml:lang="zh_TW">
          <minus xml:lang="zh_TW" />
          <apply xml:lang="zh_TW">
            <mult xml:lang="zh_TW" />
            <real xml:lang="zh_TW">17</real>
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">e</id>
              <real xml:lang="zh_TW">5</real>
            </apply>
          </apply>
          <real xml:lang="zh_TW">2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" />
    </resultFormat>
  </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：

按一下以複製此運算式

按一下以複製此運算式

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  <math resultRef="17" xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <limit xml:lang="zh_TW" direction="right" />
      <lambda xml:lang="zh_TW">
        <boundVars xml:lang="zh_TW">
          <id xml:space="preserve" xml:lang="zh_TW">ε</id>
        </boundVars>
        <parens xml:lang="zh_TW">
          <apply xml:lang="zh_TW">
            <plus xml:lang="zh_TW" />
            <apply xml:lang="zh_TW">
              <integral xml:lang="zh_TW" />
              <lambda xml:lang="zh_TW">
                <boundVars xml:lang="zh_TW">
                  <id xml:space="preserve" xml:lang="zh_TW">x</id>
                </boundVars>
                <apply xml:lang="zh_TW">
                  <id xml:space="preserve" xml:lang="zh_TW">f</id>
                  <id xml:space="preserve" xml:lang="zh_TW">x</id>
                </apply>
              </lambda>
              <lowerBound xml:lang="zh_TW">
                <id xml:space="preserve" xml:lang="zh_TW">a</id>
              </lowerBound>
              <upperBound xml:lang="zh_TW">
                <apply xml:lang="zh_TW">
                  <minus xml:lang="zh_TW" />
                  <id xml:space="preserve" xml:lang="zh_TW">c</id>
                  <id xml:space="preserve" xml:lang="zh_TW">ε</id>
                </apply>
              </upperBound>
            </apply>
            <apply xml:lang="zh_TW">
              <integral xml:lang="zh_TW" />
              <lambda xml:lang="zh_TW">
                <boundVars xml:lang="zh_TW">
                  <id xml:space="preserve" xml:lang="zh_TW">x</id>
                </boundVars>
                <apply xml:lang="zh_TW">
                  <id xml:space="preserve" xml:lang="zh_TW">f</id>
                  <id xml:space="preserve" xml:lang="zh_TW">x</id>
                </apply>
              </lambda>
              <lowerBound xml:lang="zh_TW">
                <apply xml:lang="zh_TW">
                  <plus xml:lang="zh_TW" />
                  <id xml:space="preserve" xml:lang="zh_TW">c</id>
                  <id xml:space="preserve" xml:lang="zh_TW">ε</id>
                </apply>
              </lowerBound>
              <upperBound xml:lang="zh_TW">
                <id xml:space="preserve" xml:lang="zh_TW">b</id>
              </upperBound>
            </apply>
          </apply>
        </parens>
      </lambda>
      <real xml:lang="zh_TW">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="19" xml:lang="zh_TW">
    <symEval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <div xml:lang="zh_TW" />
            <real xml:lang="zh_TW">1</real>
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
              <real xml:lang="zh_TW">3</real>
            </apply>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">-1</real>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">1</real>
        </upperBound>
      </apply>
      <command xml:lang="zh_TW">
        <placeholder xml:lang="zh_TW" />
      </command>
      <symResult xml:lang="zh_TW">
        <id labels="*" xml:space="preserve" xml:lang="zh_TW">undefined</id>
      </symResult>
    </symEval>
  </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

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  <math resultRef="21" xml:lang="zh_TW">
    <symEval xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <div xml:lang="zh_TW" />
            <real xml:lang="zh_TW">1</real>
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
              <real xml:lang="zh_TW">3</real>
            </apply>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">-1</real>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <real xml:lang="zh_TW">1</real>
        </upperBound>
      </apply>
      <command xml:lang="zh_TW">
        <id xml:space="preserve" xml:lang="zh_TW">cauchy</id>
      </command>
      <symResult xml:lang="zh_TW">
        <real xml:lang="zh_TW">0</real>
      </symResult>
    </symEval>
    <resultFormat xml:lang="zh_TW">
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="zh_TW" />
    </resultFormat>
  </math>
</region>

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。