# Fundamental Theorem of Calculus (FTC)

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

### Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart 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 CalculusPart I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\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.

### Bloom’s Taxonomy

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.

### Power Rule for Antiderivatives

#### Investigation 13

1. Compute the derivative $\frac{d}{dx}{x}^{n}$$\displaystyle \frac{d}{dx} x^n$

2. Then what is the antiderivative of $n{x}^{n-1}$$nx^{n-1}$, i.e., evaluate $\int n{x}^{n-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{nx^{n-1}\,dx}$? Explain.

#### Definition: Power Rule for Antiderivatives

$\int {x}^{n}\phantom{\rule{0.167em}{0ex}}dx=\frac{{x}^{n+1}}{n+1},\phantom{\rule{0.167em}{0ex}}n\ne -1$$\int{x^n\,dx}=\frac{x^{n+1}}{n+1},\,n\ne-1$

#### Derivation of Power Rule

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

So, we know how to evaluate $\int n{x}^{n-1}\phantom{\rule{0.167em}{0ex}}dx$$\int{nx^{n-1}\,dx}$ by using a known derivative formula.

Now, let's consider the following integral: $\int {x}^{n}\phantom{\rule{0.167em}{0ex}}dx$$\int{x^{n}\,dx}$.

We know $\frac{d}{dx}{x}^{n+1}=\left(n+1\right){x}^{n}$$\frac{d}{dx} x^{n+1}=(n+1)x^{n}$.

Rewrite this equation by dividing both sides of the the equation by $\left(n+1\right)$$(n+1)$:

$\frac{d}{dx}{x}^{n+1}=\left(n+1\right){x}^{n}$$\frac{d}{dx} x^{n+1}=(n+1)x^{n}$

$⇒\left(\frac{1}{n+1}\right)\frac{d}{dx}{x}^{n+1}=\frac{\left(n+1\right){x}^{n}}{n+1}$$\Rightarrow \left(\frac{1}{n+1}\right)\frac{d}{dx} x^{n+1}=\frac{(n+1)x^{n}}{n+1}$

$⇒\frac{d}{dx}\left(\frac{{x}^{n+1}}{n+1}\right)={x}^{n}$$\Rightarrow \frac{d}{dx} \left(\frac{x^{n+1}}{n+1}\right)=x^{n}$

$⇒\frac{{x}^{n+1}}{n+1}+C=\int {x}^{n}\phantom{\rule{0.167em}{0ex}}\text{d}x$$\Rightarrow \frac{x^{n+1}}{n+1}+C=\int{x^{n}\,\text{d}x}$

#### Investigation 14

1. For the integral $\int {x}^{n}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int{x^n\,dx}$, explain why $n\ne -1$$n\ne -1$.

2. Evaluate: $\int {x}^{-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int{x^{-1}\,dx}$. Explain.

#### Investigation 15

Using your Calculus I knowledge, explain why $\left(\frac{1}{n+1}\right)\frac{d}{dx}{x}^{n+1}=\frac{d}{dx}\left(\frac{{x}^{n+1}}{n+1}\right)$$\displaystyle\left(\frac{1}{n+1}\right)\frac{d}{dx} x^{n+1}= \frac{d}{dx} \left(\frac{x^{n+1}}{n+1}\right)$