Table of Contents

Also see the outline of the entire book as planned, including draft chapters that are not yet completed.

• Matematika
  Adaptasi Bahasa Indonesia dari Algebrica
• Part 1: Sets and numbers
  • Chapter 1. Sets
    A set is a collection of objects called elements that are considered as a whole. They are represented by uppercase letters A, B, C, and their elements by lowercase letters.
  • Chapter 2. Types Of Numbers
    Numbers organized into nested families, each extending the previous one to accommodate quantities that the smaller family cannot represent.
  • Chapter 3. Natural Numbers
    The natural numbers arise from the act of counting, but a modern treatment demands an axiomatic foundation that makes their properties explicit and independent from any informal notion.
  • Chapter 4. Integers
    Among the different types of numbers, the integers emerge when we extend the natural numbers to include the additive opposites of every positive quantity.
  • Chapter 5. Modulo Operator
    The modulo operator is one of the most frequently used operations in integer arithmetic. Given two integers, it returns the remainder left over after dividing the first by the second.
  • Chapter 6. Real Numbers
    The real numbers are introduced as a structure characterised by a combination of algebraic and order properties.
  • Chapter 7. Properties Of Real Numbers
    Before discussing the algebraic properties of real numbers, it is essential to clarify how operations are performed in an expression.
  • Chapter 8. Absolute Value
    The absolute value of a number represents its distance from zero on the number line, without considering its sign.
  • Chapter 9. Intervals
    An interval is a subset of the real line with the property that, whenever two points belong to it, every point lying between them also belongs to it.
  • Chapter 10. Supremum and Infimum
    Although real numbers are frequently introduced via their algebraic properties, the essential distinction between \mathbb{R} and \mathbb{Q} lies in their order structure, particularly a unique...
  • Chapter 11.
  • Chapter 12. Binomial Coefficient
    The binomial coefficient denotes the number of ways to select a specific number of items k from a more extensive set of n elements, disregarding the selection order.
• Part 2: Algebraic structures
  • Chapter 13.
  • Chapter 14. Rings
    A ring is an algebraic structure that extends the notion of a group by introducing a second binary operation.
  • Chapter 15. Fields
    A field is an algebraic structure in which the operations of addition and multiplication are both fully invertible, subject to the sole exception that division by zero is excluded.
  • Chapter 16. Vector Spaces
    A vector space is an algebraic structure that formalises the idea of quantities that can be scaled and combined linearly.
• Part 3: Powers radicals and logarithms
  • Chapter 17.
  • Chapter 18. Radicals
    Radicals emerge from the problem of solving equations of the form x^{n} = a, where n \in \mathbb{N}, n \geq 2, and a \in \mathbb{R}.
  • Chapter 19. Logarithms
    If a and b are positive real numbers, where a \neq 1, the logarithm of b to the base a, denoted as log{a} ⁡ ( b ), is defined as the real number c such that a^{c} = b.
• Part 4: Complex numbers
  • Chapter 20. Complex Numbers
    Complex numbers arise to overcome the limitations of the set of real numbers \mathbb{R}, particularly the impossibility of taking even-indexed roots of negative numbers.
  • Chapter 21. Operations with Complex Numbers
    A complex number z is an expression of the form z = a + b i, where a and b are real numbers and i is the imaginary unit, characterized by the defining relation i^{2} = - 1.
  • Chapter 22. Complex Numbers in Trigonometric Form
    The algebraic form z = a + b i represents a complex number through its real and imaginary components directly.
  • Chapter 23.
  • Chapter 24. De Moivre’s Theorem
    Suppose we want to compute the power of a complex number z \in \mathbb{C}. The most straightforward approach is to start from its algebraic form and expand the expression directly.
  • Chapter 25. Roots of Unity
    Given a positive integer n, a root of unity of order n is a complex number z satisfying the equation z^{n} = 1
• Part 5: Trigonometry
  • Chapter 26. Unit Circle
    The unit circle (or the trigonometric circle) is a circle of radius one centered at the origin of the Cartesian plane.
  • Chapter 27. Sine and Cosine
    Sine and cosine are the two primary trigonometric functions. Given an oriented angle \theta, represented on the unit circle by a point P, the sine and cosine of \theta are defined respectively as the...
  • Chapter 28. Tangent and Cotangent
    Tangent and cotangent are two trigonometric ratios derived from sine and cosine. Given an oriented angle \theta, the tangent is defined as the ratio of the sine of \theta to its cosine, and the...
  • Chapter 29. Secant and Cosecant
    Consider the unit circle centered at the origin \text{O} = ( 0 , 0 ) with radius 1.
  • Chapter 30. Arcsine and Arccosine
    The arcsine is the inverse of the sine function. Given a number x \in [ - 1 , 1 ] (i.e., the range of values the sine function can attain), arcsin ⁡ ( x
  • Chapter 31. Arctangent and Arccotangent
    In the unit circle, the tangent of an angle \theta can be visualized as the length of the segment tangent to the circle at the point where the terminal side meets it, measured along the vertical...
  • Chapter 32. Hyperbolic Sine and Cosine
    We have seen that the sine of an angle can be introduced geometrically by looking at how a point moves along the unit circle.
  • Chapter 33. Hyperbolic Tangent and Cotangent
    The hyperbolic tangent and cotangent arise from the hyperbolic sine and cosine in exactly the same way that the circular tangent and cotangent arise from the circular sine and cosine.
  • Chapter 34.
  • Chapter 35. Trigonometric Identities
    A trigonometric identity is an equation involving trigonometric functions that holds for every admissible value of the variables.
  • Chapter 36. Pythagorean Identity
    The Pythagorean identity is an equation that connects trigonometry and geometry, and it derives directly from the Pythagorean theorem, which relates the sides of a right triangle.
  • Chapter 37.
  • Chapter 38. Reduction Formulas and Reference Angles
    Given an angle \theta in standard position on the unit circle, the acute angle formed between its terminal side and the horizontal axis is called the reference angle of \theta, and is usually denoted...
  • Chapter 39. The Law of Sines
    The law of sines states that in any triangle, the ratio between the length of a side and the sine of its opposite angle is the same for all three sides.
  • Chapter 40. The Law of Cosines
    The law of cosines relates the sides of any triangle through the angle opposite to one of them. It can be viewed as a generalisation of the Pythagorean theorem, valid not only for right triangles but...
• Part 6: Polynomials
  • Chapter 41.
  • Chapter 42. Monomials
    A monomial is an algebraic expression consisting of a single term. It is written as the product of a numerical coefficient and one or more variables, each raised to a non-negative integer exponent.
  • Chapter 43. Binomials
    A binomial refers to a polynomial that contains exactly two non-zero terms. Its general form is expressed as ( a + b ) or ( a - b ).
  • Chapter 44. Trinomials
    A trinomial is defined as a polynomial consisting of exactly three non-zero, pairwise distinct terms. More generally, within a commutative ring with unity, a trinomial in the indeterminate x is any...
  • Chapter 45. Adding and Subtracting Polynomials
    Let R be a commutative ring and R [ x ] the ring of polynomials in one indeterminate over R. Consider two polynomials P ( x ) and Q (
  • Chapter 46. Polynomial Division
    Let P ( x ) and D ( x ) be polynomials in \mathbb{R} [ x ] with D ( x ) \neq 0.
  • Chapter 47. The Synthetic Division Method
    The synthetic division (or Ruffini’s rule) is a method for dividing a polynomial by a binomial of the form ( x - a ).
  • Chapter 48. Roots of a Polynomial
    Let p ( x ) be a polynomial with coefficients in a field \mathbb{F}, typically \mathbb{R} or \mathbb{C}. A root or zero of p is any element r \in \mathbb{F} such that
  • Chapter 49. Binomial Theorem
    The binomial theorem asserts that for any positive integer n, the expression ( a + b )^{n} can be expanded as a finite sum of n + 1 terms.
  • Chapter 50.
  • Chapter 51.
  • Chapter 52. Notable Products
    Notable products are identities describing the expansion or factorisation of polynomials such as binomials or trinomials.
  • Chapter 53. Partial Fraction Decomposition
    Partial fraction decomposition is a practical and widely used method for rewriting a rational function (a quotient of two polynomials) as a sum of simpler elementary fractions.
• Part 7: Equations
  • Chapter 54. Equations
    An equation is a mathematical statement asserting that two expressions take the same value, typically written in the form F ( x ) = G ( x
  • Chapter 55.
  • Chapter 56. Linear Equations
    Linear equations describe the relationship between variables linearly. These equations are considered the simplest form of equations involving addition, subtraction, and multiplication.
  • Chapter 57.
  • Chapter 58.
  • Chapter 59. Factoring Quadratic Equations
    A quadratic equation is a type of polynomial expression that consists of one or more terms, where each term is a product of a constant coefficient, a variable raised to a power, and a possible...
  • Chapter 60. Incomplete Quadratic Equations
    A quadratic equation is considered incomplete if it lacks one of the terms from the standard form a x 2 + b x + c = 0, as long as the x^{2} term is present.
  • Chapter 61. The Geometric Interpretation of Quadratic Equations
    The graphical representation of the related function y = a x^{2} + b x + c, associated with a quadratic equation a x^{2} + b x + c = 0, is a parabola.
  • Chapter 62. Loss of Roots
    Loss of roots occurs when an algebraic manipulation eliminates one or more roots of an equation, yielding a result that is only a partial solution set.
  • Chapter 63.
  • Chapter 64.
  • Chapter 65. Trinomial Equations
    Trinomial equations are a specific type of polynomial equation that consist of three terms involving constants and powers of a variable.
  • Chapter 66. Rational Equations
    Rational equations feature at least one fraction in which the numerator and denominator are polynomials.
  • Chapter 67. Irrational Equations
    Irrational equations, also known as radical equations, are equations in which the unknown variable x appears within a radical or is represented by a fractional exponent.
  • Chapter 68. Absolute Value Equations
    Absolute value equations are a particular class of equations in which the variable x appears within an absolute value expression.
  • Chapter 69.
  • Chapter 70. Logarithmic Equations
    Logarithmic equations are equations in which the unknown appears inside a logarithm. To solve them, it is crucial to understand the properties of logarithms and how these can be applied to isolate...
  • Chapter 71. Homogeneous Trigonometric Equations
    Homogeneous trigonometric equations are equations in which all terms involve trigonometric functions, such as sine and cosine, raised to the same degree.
  • Chapter 72.
  • Chapter 73. Equations with Parameters
    A parametric equation refers to a family of equations indexed by a real quantity that is allowed to vary freely.
  • Chapter 74. Linear Equations with Parameters
    A first-degree linear equation involving parameters is an equation in which the unknown variable appears only to the first power, while some of the coefficients are represented by symbolic quantities...
  • Chapter 75. Quadratic Equations with Parameters
    Quadratic equations are usually introduced in their classical form, where all coefficients are fixed real numbers.
• Part 8: Inequalities
  • Chapter 76. Linear Inequalities
    An inequality is a mathematical statement involving algebraic expressions for which we seek the values of the variables that make the inequality true.
  • Chapter 77. Quadratic Inequalities
    A quadratic inequality or inequality of degree two is a second-degree polynomial inequality in one variable.
  • Chapter 78. Sign Analysis in Inequalities
    Sign analysis of inequalities is a method for determining the intervals in which a given expression is positive, negative, or zero.
  • Chapter 79. Rational Inequalities
    A rational inequality is an inequality that involves at least one rational expression, that is, a ratio in which both the numerator and the denominator are polynomials.
  • Chapter 80. Irrational Inequalities
    An irrational inequality is an inequality in which the unknown appears under a radical sign or is raised to a fractional exponent. More precisely, it is an inequality of the form F (
  • Chapter 81.
  • Chapter 82. Logarithmic Inequalities
    Logarithmic inequalities are inequalities that involve one or more logarithmic expressions, in which the unknown x appears either in the argument of the logarithm or, in some cases, in the base...
  • Chapter 83. Trigonometric Inequalities
    A trigonometric inequality is an inequality in which the unknown appears as the argument of one or more trigonometric functions.
  • Chapter 84. Systems of Inequalities
    A system of inequalities is a collection of two or more inequalities that involve one or several variables and are considered simultaneously.
• Part 9: Lines planes and conic sections
  • Chapter 85. Lines
    A line is a fundamental geometric object made up of infinitely many points aligned in a perfectly straight path. It has no thickness, no endpoints, and it extends endlessly in both directions.
  • Chapter 86.
  • Chapter 87. Polar Coordinates
    The Cartesian coordinate system describes a point in the plane by projecting it onto two perpendicular axes. This representation privileges horizontal and vertical directions.
  • Chapter 88. Parabola
    When a plane intersects a cone, the shape formed at the intersection, when projected onto the plane, can be a circumference, a parabola, an ellipse, or a hyperbola.
  • Chapter 89.
  • Chapter 90. Ellipse
    When introducing the parabola, we saw that when a plane intersects a cone, the resulting shape, when projected onto the plane, can be a circumference, a parabola, an ellipse, or a hyperbola.
  • Chapter 91. Hyperbola
    When introducing the parabola, we saw that when a plane intersects a cone, the resulting shape, when projected onto the plane, can be a circumference, a parabola, an ellipse, or a hyperbola.
• Part 10: Vectors and matrices
  • Chapter 92. Vectors and Matrices
    A vector is a quantity characterised by both a magnitude and a direction, in contrast to a scalar, which is described by magnitude alone.
  • Chapter 93. Vectors and Matrices
    A matrix is a rectangular array of real numbers arranged in rows and columns. A matrix with m rows and n columns is said to have dimensions m \times n, and is called an m \times n matrix.
  • Chapter 94. Vectors and Matrices
    To every square matrix of order n one can associate a real number called the determinant of the matrix, denoted det ( A ) or | A |.
  • Chapter 95.
  • Chapter 96. Vectors and Matrices
    The rank of a matrix A, denoted r ( A ) or rank ( A ), is the maximum number of linearly independent rows (or equivalently, columns) of A.
  • Chapter 97. Vectors and Matrices
    In linear algebra, a linear combination is the fundamental operation that relates vectors to one another within a given collection.
  • Chapter 98. Vectors and Matrices
    A linear transformation, represented by a square matrix A, acts on vectors by moving them in space. It can stretch, compress, rotate, or reflect them, and in general the image of a vector points in a...
  • Chapter 99. Vectors and Matrices
    A square matrix is said to be diagonalizable when it is possible to find a basis of the underlying vector space consisting entirely of eigenvectors of that matrix.
• Part 11: Linear systems
  • Chapter 100. Systems of Linear Equations
    Linear systems model problems where multiple conditions must be satisfied at the same time. They form the basis of many solution methods in algebra and applied mathematics.
  • Chapter 101. Cramer’s Rule
    Cramer’s Rule provides a method for solving systems of n linear equations in n unknowns, by using the determinant of the system’s coefficient matrix.
  • Chapter 102. Gaussian Elimination
    The Gauss method, or Gaussian elimination, is a technique used to solve systems of n linear equations in n unknowns.
• Part 12: Sequences
  • Chapter 103. Principle of Mathematical Induction
    Mathematical induction is a fundamental principle used to rigorously prove statements concerning natural numbers.
  • Chapter 104. Sequences
    A sequence is an ordered collection of elements, each assigned to a specific position indexed by a natural number. Let us consider the set of real numbers \mathbb{R}.
  • Chapter 105. Convergent and Divergent Sequences
    We introduced sequences as an ordered collection of elements, each assigned to a specific position indexed by a natural number. To every sequence ( a{n}
  • Chapter 106.
  • Chapter 107. Arithmetic Sequence
    A sequence a{n} is called an arithmetic sequence (or arithmetic progression) if it consists of numbers arranged in such a way that the difference between any term and the one before it is constant.
  • Chapter 108. Geometric Sequence
    A sequence a{n} is called a geometric sequence (or geometric progression) if it consists of numbers arranged in such a way that the ratio between any term and the one before it is constant.
  • Chapter 109. Cauchy Sequence
    A Cauchy sequence is a special type of sequence where, as you move further along, the terms get closer and closer to each other.
  • Chapter 110.
  • Chapter 111. Sequences of Functions
    Imagine you have a list of different functions, where each function in the list is linked to a number n = 1 , 2 , 3 \ldots \in \mathbb{N}.
• Part 13: Series
  • Chapter 112. Series
    The concept of a series is closely tied to infinite sequences of real numbers, with the main goal of studying the behavior of their infinite sum.
  • Chapter 113. Cauchy’s Convergence Criterion for Series
    Cauchy’s criterion is a useful tool for proving that a series converges without needing to know its sum.
  • Chapter 114.
  • Chapter 115.
  • Chapter 116.
  • Chapter 117. Integral Test for Series Convergence
    Determining the sum of an infinite series and assessing its convergence or divergence is not always straightforward.
  • Chapter 118. Root Test for Series Convergence
    The root test is a method used to determine whether an infinite series converges or diverges. It is particularly useful when each term of the series involves an expression raised to the n-th power,...
  • Chapter 119. Leibniz’s Criterion
    Leibniz’s criterion is used to study the convergence of alternating series, those composed of an infinite sequence of positive and negative terms that alternate in sign.
  • Chapter 120. Function Series
  • Chapter 121. Power Series
  • Chapter 122. Taylor Series
    The idea behind a Taylor series is to replace a function with an infinite polynomial whose coefficients are entirely determined by the local behaviour of the function at a single point.
  • Chapter 123. Fourier Series
    A Fourier series represents a periodic function as an infinite sum of sine and cosine functions. More precisely, it shows that periodic behavior can be decomposed into elementary harmonic...
• Part 14: Functions
  • Chapter 124. Functions
    A function is a mathematical rule that connects two non-empty subsets of the real numbers, typically denoted as A \subseteq \mathbb{R} and B \subseteq \mathbb{R}.
  • Chapter 125. Determining the Domain of a Function
    In the introduction to functions, we discussed the idea of the domain of a function, that is, the set of input values for which an expression is mathematically meaningful.
  • Chapter 126. Even and Odd Functions
    When analyzing the behavior of a function, it is useful to investigate whether the function exhibits symmetry with respect to the coordinate axes.
  • Chapter 127.
  • Chapter 128. Convexity and Concavity of Functions
    The study of a function’s behaviour involves not only determining where it increases or decreases, but also understanding how its graph bends within an interval.
  • Chapter 129. Composite Functions
    When we talk about composite functions, we refer to the process of applying one function to the result of another. In other words, given two functions f ( x
  • Chapter 130.
  • Chapter 131. Continuous Functions
    The concept of continuity of a function is used to determine whether the function behaves predictably near a point, without jumps, holes, or abrupt changes. Formally, a function y = f (
  • Chapter 132. Uniform Continuity
    Ordinary continuity describes the local behaviour of a function, where small changes in the input near each point result in small changes in the output.
  • Chapter 133. Discontinuities of Real Functions
    Continuity is a property of a function in which small variations in the input result in correspondingly small variations in the output within the neighbourhood of a given point.
  • Chapter 134. Analyzing the Graphs of Functions
    Analyzing the graph of a function y = f ( x ) allows us to analyze its behavior and key characteristics, providing valuable insights into its mathematical properties.
  • Chapter 135.
  • Chapter 136. Rational Functions
    Rational functions are functions in which both the numerator and the denominator are polynomials, typically of degrees n and m. They are usually written in the form
  • Chapter 137. Logarithmic Function
    The logarithmic function is the inverse of the exponential function. Therefore, its domain and range are inverted compared to the exponential function.
  • Chapter 138.
  • Chapter 139.
  • Chapter 140.
  • Chapter 141. Sine Function
    The sine function f ( x ) = sin ⁡ ( x ) assigns to each angle x, expressed in radians, its corresponding sine value.
  • Chapter 142. Cosine Function
    The cosine function f ( x ) = cos ⁡ ( x ) assigns to each angle x, expressed in radians, its corresponding cosine value.
  • Chapter 143. Tangent Function
    The tangent function f ( x ) = tan ⁡ ( x ) assigns to each angle x, expressed in radians, its corresponding tangent value.
  • Chapter 144. Cotangent Function
    The cotangent function f ( x ) = cot ⁡ ( x ) assigns to each angle x, expressed in radians, its corresponding cotangent value.
  • Chapter 145. Secant Function
    The secant function f ( x ) = sec ⁡ ( x ) is defined as the reciprocal of the cosine function.
  • Chapter 146. Cosecant Function
    The cosecant function f ( x ) = csc ⁡ ( x ) is defined as the reciprocal of the sine function.
  • Chapter 147.
  • Chapter 148. Sigmoid Function
    The sigmoid function is a real-valued function of a real variable that takes values strictly between 0 and 1, approaching each of the two extremes asymptotically.
• Part 15: Limits
  • Chapter 149. Limits
    The concept of a limit is fundamental in mathematics. Intuitively, the limit of a function f ( x
  • Chapter 150. Algebra of Limits
    The definition of a limit offers a framework for describing how a function f ( x ) approaches a specific value near a given point x{0}.
  • Chapter 151. Squeeze Theorem
    The Squeeze Theorem, also referred to as the Sandwich Theorem, provides a method for determining the limit of a function when direct evaluation is challenging or when the function displays complex...
  • Chapter 152. Remarkable Limits
    Notable limits play a central role in mathematical analysis. They are repeatedly used in calculations and help describe both the local behaviour of functions and their behaviour at infinity.
  • Chapter 153. Asymptotes
    Asymptotes are a fundamental concept in mathematical analysis. They are lines that a function approaches indefinitely without ever reaching, which helps to characterise the function’s behaviour,...
  • Chapter 154.
  • Chapter 155. Little-o Notation
    The symbol o ( x ), referred to as little-o of x, belongs to the Landau symbol family, which is used to characterise asymptotic relationships between functions.
  • Chapter 156. Big O Notation
    The symbol O ( x ), commonly known as big O of x, is part of the Landau symbol family.
• Part 16: Derivatives
  • Chapter 157. Difference Quotient
    Consider a function y = f ( x ) defined on the interval [ a , b ]
  • Chapter 158. Derivatives
    Consider a function y = f ( x ) defined on an interval [ a , b ]. The derivative of f at a point c \in ( a , b
  • Chapter 159. Derivative of a Composite Function
    Let g be differentiable at x, and let f be differentiable at z = g ( x ). Then the composite function y = f ( g ( x )
  • Chapter 160. Non-Differentiable Points
    In the entry on derivatives, we saw that if a function f ( x ) is differentiable at a point c, then the function is continuous at that point.
  • Chapter 161. Differential of a Function
    Consider f ( x ) a differentiable function on the interval [ a , b ]. Since the function is differentiable, it is also continuous on the given interval.
  • Chapter 162. Derivative of Composite Power Functions
    We have previously introduced how to calculate the derivative of a function at a point using the definition of the difference quotient.
  • Chapter 163. Maximum, Minimum, and Inflection Points
    The maximum and minimum of a function f ( x ) represent, respectively, the highest and lowest values that the function can attain within its domain.
  • Chapter 164. Partial Derivatives
    Partial derivatives generalise the concept of the derivative to functions of several real variables. For a function of a single variable, the derivative quantifies the rate of change of the function...
• Part 17: Differential calculus theorems
  • Chapter 165.
  • Chapter 166. Fermat’s Theorem
    Fermat’s Theorem states that any relative maximum or minimum of a differentiable function within its domain must occur at a stationary point, that is, a point where the first derivative is equal to...
  • Chapter 167.
  • Chapter 168. Lagrange’s Theorem
    The Lagrange’s theorem, also known as the mean value theorem, states the following. Consider a function f ( x ), continuous in the closed and bounded interval [
  • Chapter 169. Cauchy’s Theorem
    Cauchy’s Theorem establishes a relationship between the changes of two functions over a given interval. Specifically, if f ( x ) and g ( x
  • Chapter 170. L’Hôpital’s Rule
    L’Hôpital’s rule is a method for evaluating certain limits that result in indeterminate forms. The theorem establishes a criterion for resolving the indeterminate form of the limit of one or more...
• Part 18: Integrals
  • Chapter 171. Indefinite Integrals
    Differentiation assigns to each function a unique derivative by definition. The inverse process seeks to determine whether, for a given function f ( x
  • Chapter 172. Definite Integrals
    To introduce the concept of the definite integral, consider a function f ( x ) defined on a closed interval [ a , b ].
  • Chapter 173. Fundamental Theorem of Calculus
    The Fundamental Theorem of Calculus establishes the exact relationship between differentiation and integration. These two operations arise from different initial motivations.
  • Chapter 174. Integration by Substitution
    Integration by substitution is a technique used to simplify an integral by introducing a suitable substitution.
  • Chapter 175.
  • Chapter 176. Finding Areas by Integration
    Building on the concept of definite integrals, which measure the area between a curve and the x-axis, we can extend the same idea to find the area enclosed between two curves. Let f (
  • Chapter 177. Integral of the Exponential Function
    An exponential function is a function of the form e^{x} or \alpha^{x} (with \alpha > 0 and \alpha \neq 1).
  • Chapter 178. Integral of Trigonometric Functions
    In the section on functions, you’ll find the integrals of the main trigonometric functions (for example sine, cosine, tangent, cotangent, and the others.
  • Chapter 179.
  • Chapter 180. Trigonometric Substitution for Integrals
    Trigonometric substitution is a method for evaluating integrals that contain square roots of quadratic expressions.
  • Chapter 181. Improper Integrals
    Improper integrals are integrals in which either the interval of integration is unbounded, or the integrand becomes unbounded at one or more points, or both.
  • Chapter 182. Riemann Integrability Criteria
    The Riemann integral is built to measure the net area under a bounded function on a closed interval by approximating it with rectangles.
• Part 19: Differential equations
  • Chapter 183. Differential Equations
    An ordinary differential equation is a relation involving a function f ( x ), defined on an interval I \subset \mathbb{R}, together with a finite number of its derivatives.
• Part 20: Probability and statistics
  • Chapter 184. Probability and Statistics
    The mean is among the most fundamental tools in statistics for describing the central behavior of a data distribution.
  • Chapter 185. Probability and Statistics
    The arithmetic mean is the most common and intuitive form of average. As a special case within the broader family of power means it expresses the representative value of a data set by dividing the...
  • Chapter 186. Probability and Statistics
    The geometric mean belongs to the family of power means. Unlike the simple arithmetic mean, it is based on the product of the elements rather than their sum, making it especially useful for measuring...
  • Chapter 187. Probability and Statistics
    The harmonic mean belongs to the broader family of power means and plays a distinctive role whenever the data being analyzed combine reciprocally rather than additively.
  • Chapter 188. Root Mean Square
    The quadratic mean, also called the root mean square, belongs to the general family of power means. It is obtained by taking the square root of the arithmetic mean of the squared values in a dataset.
  • Chapter 189. Median and Quantiles
    The median is a key measure of central tendency used to describe the typical value within a dataset. While the mean expresses the numerical balance of all values, the median focuses instead on...
  • Chapter 190. Variance
    One of the key principles in statistics is that the mean alone cannot fully describe a dataset. What truly matters is understanding how the individual observations are spread around the mean, that...
  • Chapter 191. Discrete Random Variables
    A discrete random variable is a function that assigns a real number to each element of a discrete sample space.
  • Chapter 192. Continuous Random Variables
    A continuous random variable is a function that assigns a real number to each element of a continuous sample space.
  • Chapter 193. Mean or Expected Value of a Random Variable
    The mean represents a fundamental statistical measure that characterizes the central tendency of a dataset.
  • Chapter 194. Variance and Covariance of a Random Variable
    In descriptive statistics, the variance expresses how much a set of values differs, on average, from its mean.
  • Chapter 195.
  • Chapter 196. Binomial Distribution
    The binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent experiments, each one following a Bernoulli distribution with the...
  • Chapter 197. Hypergeometric Distribution
    The hypergeometric distribution is a discrete probability distribution that describes the number of successes drawn from a finite population without replacement.
  • Chapter 198. Geometric Distribution
    The geometric distribution describes the number of independent trials required to observe the first success in a repeated experiment.
  • Chapter 199. Poisson Distribution
    The Poisson distribution is a discrete probability distribution that describes how many times a specific event may occur within a fixed period of time or space.
  • Chapter 200. Uniform Distribution
    The uniform distribution is one of the simplest continuous distributions to describe. It models a random variable that can take any value within a specified interval, assigning the same probability...
  • Chapter 201. Beta Distribution
    The beta distribution is a continuous probability distribution defined over the open interval ( 0 , 1 ).
  • Chapter 202. Normal Distribution
    The normal distribution, also known as the Gaussian distribution, is one of the most important continuous probability distributions in both probability and statistics.
  • Chapter 203. Standard Normal Z Table
    A generic normal distribution \mathcal{N} ( x ; \mu , \sigma ) can always be transformed into its standardized form N ( x ; 0 , 1
  • Chapter 204. Gamma Distribution
    The gamma distribution is a continuous probability distribution defined on the positive half-line. It is used to model waiting times, event durations, and phenomena where independent contributions...
  • Chapter 205. Chi-square Distribution
    The chi-square distribution is a continuous probability distribution that arises from analyzing how the sum of squared observations behaves when those observations follow a standard normal...
  • Chapter 206. Student’s t Distribution
    In statistical inference, when the goal is to draw conclusions about the mean of a normally distributed population but the variance is unknown and must be estimated from the sample, the standard...
  • Chapter 207. Exponential Distribution
    The exponential distribution characterizes the time elapsed between random, independent events occurring at a constant average rate.
  • Chapter 208. Sampling Distributions
    A sampling distribution represents the distribution of a statistic obtained from all possible samples of a given size drawn from a population.
  • Chapter 209. Bayes’ Theorem
    Bayes’ Theorem is a fundamental result in probability theory that describes how to compute the conditional probability of a hypothesis given observed evidence.
  • Chapter 210. Confidence Intervals
    When we rely on a sample to learn something about an unknown population parameter, say the mean \mu, a natural first step is to use a point estimator.
• Part 21: Kinematics
  • Chapter 211.
  • Chapter 212.
  • Chapter 213. Simple Harmonic Motion
    A simple harmonic motion is a straight-line motion obtained by projecting the uniform circular motion of a body onto a fixed diameter of the circle.
• Part 22: Other topics
  • Chapter 214. Cosine Similarity
    When working with large collections of text, it becomes necessary to have tools that allow a computer to evaluate how similar two documents are to each other.
  • Chapter 215. Propositional Logic
    Propositional logic studies the inferential relationships among sentences, focusing on a class of logical operators known as propositional connectives.
  • Chapter 216. The Backpropagation Algorithm
    Training a neural network means finding the values of the parameters, typically weight matrices, that minimize a loss function with respect to the training data.
• References