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Calculus 2 Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Evaluating Integrals

Term		Definition		Formula		Example
Integration by Parts	A technique used to integrate the product of two functions by transforming it into a simpler form.	$\int u d v = u v - \int v d u$ where $u$ and $v$ are differentiable functions of $x$ .	1. Consider the integral $\int x \sin (x) d x$ $u = x, d v = \sin (x) d x$ $d u = d x, v = - \cos (x)$ 2. Applying the integration by parts formula: $\int x \sin (x) d x = - x \cos (x) - \int - \cos (x) d x$ $= - x \cos (x) + \sin (x) + C$
Trigonometric Substitution	A technique used to simplify integrals involving square roots of quadratic expressions using trigonometric identities.		Consider the integral $\int \sqrt{4 - x^{2}} d x$ . We can use trigonometric substitution with $x = 2 \sin (θ)$ : $d x = 2 \cos (θ) d θ$ $\sqrt{4 - x^{2}} = \sqrt{4 - (2 \sin (θ))^{2}} = 2 \cos (θ)$ The integral becomes: $\int 2 \cos^{2} (θ) d θ$
Partial Fraction Decomposition	A method used to decompose a rational function into the sum of simpler fractions, making it easier to integrate.		Consider the rational function $\frac{2 x + 3}{x^{2} + 3 x + 2}$ : The denominator factors as $(x + 1) (x + 2)$ , so we write: $\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{A}{x + 1} + \frac{B}{x + 2}$ Multiplying both sides by $x^{2} + 3 x + 2$ and simplifying, we find the values of $A$ and $B$ . $\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{A}{x + 1} + \frac{B}{x + 2}$ Multiplying both sides by $(x + 1) (x + 2)$ to clear the denominators, we get: $2 x + 3 = A (x + 2) + B (x + 1)$ Expanding and collecting like terms: $2 x + 3 = A x + 2 A + B x + B$ For $x$ terms: $A + B = 2$ For constant terms: $2 A + B = 3$ Solving these equations, we find $A = 1$ and $B = 1$ . Therefore, the decomposition is $\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{1}{x + 1} + \frac{1}{x + 2}$

Term

Definition

Formula

Example

Integration by Parts

A technique used to integrate the product of two functions by transforming it into a simpler form.

\int u d v = u v - \int v d u

where

u

and

v

are differentiable functions of

x

1. Consider the integral $\int x \sin (x) d x$
$u = x, d v = \sin (x) d x$
$d u = d x, v = - \cos (x)$

2. Applying the integration by parts formula:
$\int x \sin (x) d x = - x \cos (x) - \int - \cos (x) d x$
$= - x \cos (x) + \sin (x) + C$

Trigonometric Substitution

A technique used to simplify integrals involving square roots of quadratic expressions using trigonometric identities.

Consider the integral $\int \sqrt{4 - x^{2}} d x$ .
We can use trigonometric substitution with $x = 2 \sin (θ)$ :
$d x = 2 \cos (θ) d θ$
$\sqrt{4 - x^{2}} = \sqrt{4 - (2 \sin (θ))^{2}} = 2 \cos (θ)$
The integral becomes: $\int 2 \cos^{2} (θ) d θ$

Partial Fraction Decomposition

A method used to decompose a rational function into the sum of simpler fractions, making it easier to integrate.

Consider the rational function $\frac{2 x + 3}{x^{2} + 3 x + 2}$ :

The denominator factors as $(x + 1) (x + 2)$ , so we write:

$\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{A}{x + 1} + \frac{B}{x + 2}$
Multiplying both sides by $x^{2} + 3 x + 2$ and simplifying, we find the values of $A$ and $B$ .
$\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{A}{x + 1} + \frac{B}{x + 2}$
Multiplying both sides by $(x + 1) (x + 2)$ to clear the denominators, we get:

$2 x + 3 = A (x + 2) + B (x + 1)$
Expanding and collecting like terms:

$2 x + 3 = A x + 2 A + B x + B$
For $x$ terms: $A + B = 2$

For constant terms: $2 A + B = 3$

Solving these equations, we find $A = 1$ and $B = 1$ .
Therefore, the decomposition is $\frac{2 x + 3}{x^{2} + 3 x + 2} = \frac{1}{x + 1} + \frac{1}{x + 2}$

Applications of Integration

Term	Definition	Formula
Volume of Solids of Revolution (Disk Method)	The volume of a solid formed by rotating a region bounded by a curve around a line, calculated by summing the volumes of infinitesimally thin disks along the axis of rotation.	$V = π \int_{a}^{b} [f (x)]^{2} d x$ $V = π \int_{c}^{d} [g (y)]^{2} d y$
Volume of Solids of Revolution (Shell Method)	The volume of a solid formed by rotating a region bounded by a curve around a line, calculated by summing the volumes of infinitesimally thin cylindrical shells along the axis of rotation.	$V = 2 π \int_{a}^{b} x \cdot f (x) d x$ $V = 2 π \int_{c}^{d} y \cdot g (y) d y$
Arc Length	The length of a curve, or a portion of a curve.	$L = \int_{a}^{b} \sqrt{1 + [f^{'} (x)]^{2}} d x$
Surface Area	Represents the total area of the surface of a three-dimensional object. Obtained by rotating the curve $y = f (x)$ over the interval $[a, b]$ .	$A = 2 π \int_{a}^{b} f (x) \sqrt{1 + [f^{'} (x)]^{2}} d x$

Improper Integrals

Term		Definition
Improper Integral		An integral where one or both limits of integration are infinite, or where the integrand has a vertical asymptote within the interval of integration.
Type I Improper Integrals		An improper integral where the upper or lower limit of integration is infinite.
Type II Improper Integrals		An improper integral where the integrand has a vertical asymptote within the interval of integration.
Divergent Integral		An improper integral that does not have a finite value as the limits of integration approach infinity or as they approach a point of discontinuity.
Convergent Integral		An improper integral that has a finite value as the limits of integration approach infinity or as they approach a point of discontinuity.
Comparison Test		Let $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ be two series with non-negative terms. Suppose there exists a positive real number $M$ such that $0 \leq a_{n} \leq M \cdot b_{n}$ for all $n$ (or $0 \leq b_{n} \leq M \cdot a_{n}$ for all $n$ ). If $\sum_{n = 1}^{\infty} b_{n}$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges. If $\sum_{n = 1}^{\infty} b_{n}$ diverges, then $\sum_{n = 1}^{\infty} a_{n}$ diverges.

Sequences

Term	Definition	Example
Sequence	A sequence is an ordered list of numbers denoted by ${a_{n}}$ or $(a_{n})$ , where $a_{1}, a_{2}, a_{3}, \dots$ are the terms of the sequence.
Monotonic Sequences	A sequence ${a_{n}}$ that is either entirely increasing or decreasing.	${a_{n}} = {1, 2, 3, 4, 5, \dots}$
Bounded Sequences	A sequence ${a_{n}}$ is bounded if there exist constants $M$ and $N$ such that $M \leq a_{n} \leq N$ for all $n$ .	${a_{n}} = {\frac{1}{n}}_{n = 1}^{\infty}$
Arithmetic Sequence	A sequence in which the difference between consecutive terms is constant.	${a_{n}} = {2, 5, 8, 11, 14, \dots}$
Geometric Sequence	A sequence in which the ratio of any two consecutive terms is constant.	${a_{n}} = {2, 6, 18, 54, 162, \dots}$
Fibonacci Sequence	A sequence where each term is the sum of the two preceding terms, usually starting with 0 and 1.	${a_{n}} = {0, 1, 1, 2, 3, 5, 8, 13, 21, \dots}$
Periodic Sequence	A sequence of numbers that repeats itself after a fixed number of terms, called the period.	${a_{n}} = {\sin (1), \sin (2), \sin (3), \sin (4), \dots}$
Quadratic Sequence	A sequence of numbers where each term is generated by a quadratic function of the form $a n^{2} + b n + c$ , where $a$ , $b$ , and $c$ are constants and $n$ is the term number.	${a_{n}} = {1, 4, 9, 16, 25, \dots}$
Common Ratio	A term, $a_{n} = a_{1} \cdot r^{n - 1}$ , used specifically in geometric sequences to denote the factor by which each term is multiplied to get the next term.	$a_{n} = a_{1} \cdot r^{n - 1}$
Summation Notation	A concise way to represent the sum of a sequence. It uses the symbol $\sum$ and specifies the terms to be summed, the variable of summation, and the range of values for the variable.	$\sum_{i = 1}^{n} a_{i}$

Series

Term	Definition	Formula
Series	An infinite sum of the terms of a sequence.
Divergence	Occurs when a series does not approach a finite limit as the index or the number of terms increases.
Convergence	Occurs when the terms in a series approach a finite limit as the index or the number of terms increases, indicating stability or convergence towards a specific value.
Power Series	Represents a function as a sum of powers of x. Depending on the convergence properties, a power series may converge only for certain values of x (within its radius of convergence).	$f (x) = \sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$
Radius of Convergence	Given a power series in the form $\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}$ , the radius of convergence is the number R where the series converges absolutely for all $x$ in the interval $(a - R, a + R)$ , and diverges for $x$ outside this interval.
Binomial Series	A specific form of power series representing the expansion of $(1 + x)^{n}$ . The binomial series converges when $\| x \| < 1$ .
Taylor Series	An infinite series representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.	$f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} (x - a)^{n}$
Maclaurin Series	A special case of the Taylor series where the series is centered at $x = 0$ .	$f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}$
Ratio Test	Let $\sum_{n = 1}^{\infty} a_{n}$ be a series. If $lim_{n \to \infty} \| \frac{a_{n + 1}}{a_{n}} \| < 1$ , then the series converges absolutely. If $lim_{n \to \infty} \| \frac{a_{n + 1}}{a_{n}} \| > 1$ , then the series diverges. If $lim_{n \to \infty} \| \frac{a_{n + 1}}{a_{n}} \| = 1$ , the test is inconclusive.
Root Test	Let $\sum_{n = 1}^{\infty} a_{n}$ be a series. If $lim_{n \to \infty} \sqrt[n]{\| a_{n} \|} < 1$ , then the series converges absolutely. If $lim_{n \to \infty} \sqrt[n]{\| a_{n} \|} > 1$ , then the series diverges. If $lim_{n \to \infty} \sqrt[n]{\| a_{n} \|} = 1$ , the test is inconclusive.
Integral Test	A test used to determine the convergence or divergence of an infinite series by comparing it to an improper integral. If $f (x)$ is a positive, continuous, and decreasing function for $x \geq 1$ and $a_{n} = f (n)$ for all $n$ , then the series $\sum_{n = 1}^{\infty} a_{n}$ converges if and only if the improper integral $\int_{1}^{\infty} f (x) d x$ converges.
Geometric Series	The sum of the terms in a geometric sequence.	$S = \frac{a_{1}}{1 - r}$ , where $Misplaced &$
Arithmetic Series	The sum of the terms in an arithmetic sequence.	$S = \frac{n}{2} (a_{1} + a_{n})$
Harmonic Series	A series of the form $\sum_{n = 1}^{\infty} \frac{1}{n}$ , where each term is the reciprocal of a positive integer.	$S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n} + \dots$

Parametric Equations and Polar Coordinates

Term	Symbol	Definition	Formula
Polar Coordinates	$(r, θ)$ .	Polar coordinates are a two-dimensional coordinate system where each point in the plane is specified by its distance $r$ from the origin and the angle $θ$ it makes with the positive x-axis.
Rectangular to Polar Coordinates		Involves determining the polar radius $r$ and the polar angle $θ$ using trigonometric relationships:	$r = \sqrt{x^{2} + y^{2}}$ $θ = \arctan (\frac{y}{x})$
Parametric Equations	A set of equations that express the coordinates of a point in terms of one or more independent variables, called parameters. These equations describe the motion of a point in a plane or space as functions of time or some other parameter.
Parametric Curves	Curves that are defined by parametric equations, where each coordinate of a point on the curve is given by a separate function of one or more parameters.

Common Integrals

Term		Integral
$\int 1 d x$		$x + C$
$\int x^{n} d x$		$\frac{x^{n + 1}}{n + 1} + C$ (where $n \neq - 1$ )
$\int e^{x} d x$		$e^{x} + C$
$\int \sin (x) d x$		$- \cos (x) + C$
$\int \cos (x) d x$		$\sin (x) + C$
$\int \sec^{2} (x) d x$		$\tan (x) + C$
$\int \csc^{2} (x) d x$		$- \cot (x) + C$
$\int \sec (x) \tan (x) d x$		$\sec (x) + C$
$\int \csc (x) \cot (x) d x$		$- \csc (x) + C$

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