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Antiderivatives
Источник пакета openstax.calculus_integrals · OpenStax Calculus, ch. 4 Applications of Derivatives, module m53621
- work
- urn:openstax:clir:calculus-integrals#M53621
- номер
- OpenStax Calculus, ch. 4 Applications of Derivatives, module m53621
- вид
- urn:openstax:clir:calculus-integrals#treatise
m53621/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-integrals#M53621_MAIN
- язык
- en
- официальность
- official
- материализация
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- жизненный цикл
- adopted: 2026-07-15
Публикации
| URI | Тип | documentHash | Получено |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53621/index.cnxml | application/xml; charset=utf-8 | sha256:368612823e0e54e1f08bcdddcf0c9380d69ed13e6a3dea5075e01fef655a3eb1 | 2026-09-12T00:00:00Z |
Фрагменты — 3
module/m53621#fs-id1165042478105 · defn · на него ссылаются: 7
en · official · sha256:d918628fc3a473b1f61f53afa3a0546c0c4d79c29deea6af9ebdaf7d042ee3e7EN
<note id="fs-id1165042478105">
<title>Definition</title>
<para id="fs-id1165042887564">A function <m:math><m:mi>F</m:mi></m:math> is an <term id="term-00001">antiderivative</term> of the function <m:math><m:mi>f</m:mi></m:math> if</para>
<equation id="fs-id1165042945928" class="unnumbered"><label/><m:math><m:mrow><m:msup><m:mi>F</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math></equation>
<para id="fs-id1165042964886">for all <m:math><m:mi>x</m:mi></m:math> in the domain of <m:math><m:mrow><m:mi>f</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
</note>Ссылаются нормы: InstructionAvailable, IntegrationRule, instruction_available, rule_conclusion, rule_method, rule_premise, step_justified_by
module/m53621#fs-id1165043376527 · section · на него ссылаются: 1
en · official · sha256:d5cd50df03712820760ebb3ee2c519cd8be6f22c1ec8c81afa78c792997faa7cEN
<note id="fs-id1165043376527" class="theorem">
<title>General Form of an Antiderivative</title>
<para id="fs-id1165042884523">Let <m:math><m:mi>F</m:mi></m:math> be an antiderivative of <m:math><m:mi>f</m:mi></m:math> over an interval <m:math><m:mrow><m:mi>I</m:mi><m:mo>.</m:mo></m:mrow></m:math> Then,</para>
<list id="fs-id1165043009392" list-type="enumerated" number-style="lower-roman">
<item>for each constant <m:math><m:mrow><m:mi>C</m:mi><m:mo>,</m:mo></m:mrow></m:math> the function <m:math><m:mrow><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:math> is also an antiderivative of <m:math><m:mi>f</m:mi></m:math> over <m:math><m:mrow><m:mi>I</m:mi><m:mo>;</m:mo></m:mrow></m:math></item>
<item>if <m:math><m:mi>G</m:mi></m:math> is an antiderivative of <m:math><m:mi>f</m:mi></m:math> over <m:math><m:mrow><m:mi>I</m:mi><m:mo>,</m:mo></m:mrow></m:math> there is a constant <m:math><m:mi>C</m:mi></m:math> for which <m:math><m:mrow><m:mi>G</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:math> over <m:math><m:mrow><m:mi>I</m:mi><m:mo>.</m:mo></m:mrow></m:math></item>
</list>
<para id="fs-id1165042987370">In other words, the most general form of the antiderivative of <m:math><m:mi>f</m:mi></m:math> over <m:math><m:mi>I</m:mi></m:math> is <m:math><m:mrow><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
</note>Ссылаются нормы: AntiderivativePowerJustifies
module/m53621#fs-id1165043425485 · section · на него ссылаются: 1
en · official · sha256:23e76aa66c151b780cb0b77b76deeb6be96779c5d871a16bcf57d1624404f0afEN
<note id="fs-id1165043425485" class="theorem">
<title>Properties of Indefinite Integrals</title>
<para id="fs-id1165043395041">Let <m:math><m:mi>F</m:mi></m:math> and <m:math><m:mi>G</m:mi></m:math> be antiderivatives of <m:math><m:mi>f</m:mi></m:math> and <m:math><m:mrow><m:mi>g</m:mi><m:mo>,</m:mo></m:mrow></m:math> respectively, and let <m:math><m:mi>k</m:mi></m:math> be any real number.</para>
<para id="fs-id1165043393659">Sums and Differences</para>
<equation id="fs-id1165043393663" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mtext>±</m:mtext><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>=</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mtext>±</m:mtext><m:mi>G</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></equation>
<para id="fs-id1165042328714">Constant Multiples</para>
<equation id="fs-id1165042328717" class="unnumbered"><label/><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:mo stretchy="false">∫</m:mo><m:mrow><m:mi>k</m:mi><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>x</m:mi><m:mo>=</m:mo><m:mi>k</m:mi><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>+</m:mo><m:mi>C</m:mi></m:mrow></m:mrow></m:mstyle></m:mrow></m:math></equation>
</note>Ссылаются нормы: TableAntiderivativeJustifies
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