Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomySome Additional Antiderivative RulesPower Rule for AntiderivativesInvestigation 13Definition: Power Rule for AntiderivativesDerivation of Power RuleInvestigation 14Investigation 15CC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**Objective 01–01**: I can state the*Fundamental Theorem of Calculus*.**Objective 01–02**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–03**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–04**: I can determine the properties of an area function using*FTC-I*and*Calculus I*.**Objective 01–05**: I can evaluate indefinite integrals using the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–06**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–07**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **Objective 01–08**: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.**Objective 01–09**: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Compute the derivative

$\frac{d}{dx}{x}^{n}$ Then what is the antiderivative of

, i.e., evaluate$n{x}^{n-1}$ ? Explain.$\int n{x}^{n-1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$

**NOTE**: You will **not** be assessed on the derivation of the Power Rule for Antiderivatives.

So, we know how to evaluate

Now, let's consider the following integral:

We know

Rewrite this equation by dividing both sides of the the equation by

For the integral

,$\int {x}^{n}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ **explain**why .$n\ne -1$ Evaluate:

. Explain.$\int {x}^{-1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$

Using your *Calculus I* knowledge, explain why

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**Created**: Tuesday, 25 August 2020 03:20 EDT
**Last Modified**: Tuesday, 30 May 2023 – 20:13 (EDT)