# 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

## Prerequisite Knowledge

### Calculus I

#### Properties of Definite Integrals

• ${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=-{\int }_{b}^{a}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int_a^b{f(x)\,dx}=-\int_b^a{f(x)\,dx}$
• ${\int }_{a}^{a}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=0$$\displaystyle\int_a^a{f(x)\,dx}=0$
• ${\int }_{a}^{b}c\phantom{\rule{0.167em}{0ex}}dx=c\left(b-a\right)$$\displaystyle\int_a^b{c\,dx}=c(b-a)$, where $c$$c$ is any constant
• ${\int }_{a}^{b}cf\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=c{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int_a^b{cf(x)\,dx}=c\int_a^b{f(x)\,dx}$, where $c$$c$ is any constant
• ${\int }_{a}^{b}\left[f\left(x\right)+g\left(x\right)\right]\phantom{\rule{0.167em}{0ex}}dx={\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx+{\int }_{a}^{b}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int_a^b{\left[f(x)+g(x)\right]\,dx}=\int_a^b{f(x)\,dx}+\int_a^b{g(x)\,dx}$
• ${\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]\phantom{\rule{0.167em}{0ex}}dx={\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx-{\int }_{a}^{b}g\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int_a^b{\left[f(x)-g(x)\right]\,dx}=\int_a^b{f(x)\,dx}-\int_a^b{g(x)\,dx}$
• ${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx={\int }_{a}^{c}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx+{\int }_{c}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle\int_a^b{f(x)\,dx}=\int_a^c{f(x)\,dx}+\int_c^b{f(x)\,dx}$