Key Equations

Properties of Sigma Notation	$\sum_{i = 1}^{n} c = n c$ $\sum_{i = 1}^{n} c a_{i} = c \sum_{i = 1}^{n} a_{i}$ $\sum_{i = 1}^{n} (a_{i} + b_{i}) = \sum_{i = 1}^{n} a_{i} + \sum_{i = 1}^{n} b_{i}$ $\sum_{i = 1}^{n} (a_{i} - b_{i}) = \sum_{i = 1}^{n} a_{i} - \sum_{i = 1}^{n} b_{i}$ $\sum_{i = 1}^{n} a_{i} = \sum_{i = 1}^{m} a_{i} + \sum_{i = m + 1}^{n} a_{i}$
Sums and Powers of Integers	$\sum_{i = 1}^{n} i = 1 + 2 + ⋯ + n = \frac{n (n + 1)}{2}$ $\sum_{i = 1}^{n} i^{2} = 1^{2} + 2^{2} + ⋯ + n^{2} = \frac{n (n + 1) (2 n + 1)}{6}$ $\sum_{i = 0}^{n} i^{3} = 1^{3} + 2^{3} + ⋯ + n^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$
Left-Endpoint Approximation	$A \approx L_{n} = f (x_{0}) Δ x + f (x_{1}) Δ x + ⋯ + f (x_{n - 1}) Δ x = \sum_{i = 1}^{n} f (x_{i - 1}) Δ x$
Right-Endpoint Approximation	$A \approx R_{n} = f (x_{1}) Δ x + f (x_{2}) Δ x + ⋯ + f (x_{n}) Δ x = \sum_{i = 1}^{n} f (x_{i}) Δ x$

Definite Integral	$\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x$
Properties of the Definite Integral	$\int_{a}^{a} f (x) d x = 0$ $\int_{b}^{a} f (x) d x = − \int_{a}^{b} f (x) d x$ $\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$ $\int_{a}^{b} [f (x) - g (x)] d x = \int_{a}^{b} f (x) d x - \int_{a}^{b} g (x) d x$ $\int_{a}^{b} c f (x) d x = c \int_{a}^{b} f (x)$ for constant c $\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x$

Mean Value Theorem for Integrals	If $f (x)$ is continuous over an interval $[a, b],$ then there is at least one point $c \in [a, b]$ such that $f (c) = \frac{1}{b - a} \int_{a}^{b} f (x) d x .$
Fundamental Theorem of Calculus Part 1	If $f (x)$ is continuous over an interval $[a, b],$ and the function $F (x)$ is defined by $F (x) = \int_{a}^{x} f (t) d t,$ then $F^{'} (x) = f (x) .$
Fundamental Theorem of Calculus Part 2	If f is continuous over the interval $[a, b]$ and $F (x)$ is any antiderivative of $f (x),$ then $\int_{a}^{b} f (x) d x = F (b) - F (a) .$

Net Change Theorem

F (b) = F (a) + \int_{a}^{b} F' (x) d x

\int_{a}^{b} F' (x) d x = F (b) - F (a)

Substitution with Indefinite Integrals	$\int f [g (x)] g^{'} (x) d x = \int f (u) d u = F (u) + C = F (g (x)) + C$
Substitution with Definite Integrals	$\int_{a}^{b} f (g (x)) g' (x) d x = \int_{g (a)}^{g (b)} f (u) d u$

Integrals of Exponential Functions	$\int e^{x} d x = e^{x} + C$ $\int a^{x} d x = \frac{a^{x}}{ln a} + C$
Integration Formulas Involving Logarithmic Functions	$\int x^{−1} d x = ln \| x \| + C$ $\int ln x d x = x ln x - x + C = x (ln x - 1) + C$ $\int {log}_{a} x d x = \frac{x}{ln a} (ln x - 1) + C$

Integrals That Produce Inverse Trigonometric Functions

\int \frac{d u}{\sqrt{a^{2} - u^{2}}} = {sin}^{−1} (\frac{u}{a}) + C

\int \frac{d u}{a^{2} + u^{2}} = \frac{1}{a} {tan}^{−1} (\frac{u}{a}) + C

\int \frac{d u}{u \sqrt{u^{2} - a^{2}}} = \frac{1}{a} {sec}^{−1} (\frac{u}{a}) + C