Key Equations

Key Equations

Properties of Sigma Notation	${\displaystyle \sum _{i=1}^{n}c}=nc$ ${\displaystyle \sum _{i=1}^{n}c{a}_i}=c{\displaystyle \sum _{i=1}^n{a}_i}$ ${\displaystyle \sum _{i=1}^n\left(a_i+b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}+{\displaystyle \sum _{i=1}^n{b}_i}$ ${\displaystyle \sum _{i=1}^n\left(a_i-b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}-{\displaystyle \sum _{i=1}^n{b}_i}$ ${\displaystyle \sum _{i=1}^n{a}_i}={\displaystyle \sum _{i=1}^m{a}_i}+{\displaystyle \sum _{i=m+1}^n{a}_i}$
Sums and Powers of Integers	${\displaystyle \sum _{i=1}^{n}i}=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}$ ${\displaystyle \sum _{i=1}^n{i}^2}=1^2+2^2+\text{⋯}+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ ${\displaystyle \sum _{i=0}^n{i}^3}=1^3+2^3+\text{⋯}+n^3=\frac{n^2{\left(n+1\right)}^2}{4}$
Left-Endpoint Approximation	$A\approx L_n=f\left(x_0\right)\text{Δ}x+f\left(x_1\right)\text{Δ}x+\text{⋯}+f\left(x_{n-1}\right)\text{Δ}x={\displaystyle \sum _{i=1}^{n}f\left(x_{i-1}\right)\text{Δ}x}$
Right-Endpoint Approximation	$A\approx R_n=f\left(x_1\right)\text{Δ}x+f\left(x_2\right)\text{Δ}x+\text{⋯}+f\left(x_n\right)\text{Δ}x={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)\text{Δ}x}$

Definite Integral

${\displaystyle {\int }_a^{b}f\left(x\right)dx}=\underset{n\to \infty }{{\displaystyle \text{lim}}}{\displaystyle \sum _{i=1}^{n}f\left(x_i^{*}\right)\text{Δ}x}$

Properties of the Definite Integral

${\displaystyle {\int }_a^{a}f\left(x\right)dx=0}$ ${\displaystyle {\int }_b^{a}f\left(x\right)dx}=\text{−}{\displaystyle {\int }_a^{b}f\left(x\right)dx}$ ${\displaystyle {\int }_a^b\left[f\left(x\right)+g\left(x\right)\right]dx}={\displaystyle {\int }_a^{b}f\left(x\right)dx}+{\displaystyle {\int }_a^{b}g\left(x\right)dx}$ ${\displaystyle {\int }_a^b\left[f\left(x\right)-g\left(x\right)\right]dx={\displaystyle {\int }_a^{b}f\left(x\right)dx-{\displaystyle {\int }_a^{b}g\left(x\right)dx}}}$ ${\displaystyle {\int }_a^{b}cf\left(x\right)dx}=c{\displaystyle {\int }_a^{b}f\left(x\right)}$ for constant c ${\displaystyle {\int }_a^{b}f\left(x\right)dx}={\displaystyle {\int }_a^{c}f\left(x\right)dx}+{\displaystyle {\int }_c^{b}f\left(x\right)dx}$

Mean Value Theorem for Integrals	If $f\left(x\right)$ is continuous over an interval $\left[a,b\right],$ then there is at least one point $c\in \left[a,b\right]$ such that $f\left(c\right)=\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(x\right)dx}.$
Fundamental Theorem of Calculus Part 1	If $f\left(x\right)$ is continuous over an interval $\left[a,b\right],$ and the function $F\left(x\right)$ is defined by $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt},$ then $F^{\prime }\text{(}x)=f\left(x\right).$
Fundamental Theorem of Calculus Part 2	If f is continuous over the interval $\left[a,b\right]$ and $F\left(x\right)$ is any antiderivative of $f\left(x\right),$ then ${\displaystyle {\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)}.$

Net Change Theorem

$F\left(b\right)=F\left(a\right)+{\displaystyle {\int }_a^{b}F\text{'}\left(x\right)dx}$ or ${\displaystyle {\int }_a^{b}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right)}$

Substitution with Indefinite Integrals	${\displaystyle \int f\left[g\left(x\right)\right]g^{\prime }\text{(}x)dx={\displaystyle \int f\left(u\right)du}}=F\left(u\right)+C=F\left(g\left(x\right)\right)+C$
Substitution with Definite Integrals	${\displaystyle {\int }_a^{b}f\left(g\left(x\right)\right)g\text{'}\left(x\right)dx={\displaystyle {\int }_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)du}}$

Integrals of Exponential Functions	${\displaystyle \int e^{x}dx=e^x+C}$ ${\displaystyle \int a^{x}dx=\frac{a^x}{\text{ln}a}+C}$
Integration Formulas Involving Logarithmic Functions	${\displaystyle \int x^{\mathrm{-1}}dx=\text{ln}\left|x\right|+C}$ ${\displaystyle \int \text{ln}xdx=x\text{ln}x-x+C=x\left(\text{ln}x-1\right)+C}$ ${\displaystyle \int {\text{log}}_{a}xdx=\frac{x}{\text{ln}a}\left(\text{ln}x-1\right)+C}$

Integrals That Produce Inverse Trigonometric Functions

${\displaystyle \int \frac{du}{\sqrt{a^2-u^2}}={\text{sin}}^{\mathrm{-1}}\left(\frac{u}{a}\right)+C}$ ${\displaystyle \int \frac{du}{a^2+u^2}=\frac{1}{a}{\text{tan}}^{\mathrm{-1}}\left(\frac{u}{a}\right)+C}$ ${\displaystyle \int \frac{du}{u\sqrt{u^2-a^2}}=\frac{1}{a}{\text{sec}}^{\mathrm{-1}}\left(\frac{u}{a}\right)+C}$