table of integrals

In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i.e., F′ = f. The process of solving for antiderivatives is antidifferentiation (or indefinite integration). Finding an expression for an antiderivative is harder than calculating a derivative, and may not always be possible. Antiderivatives are related to integrals through the fundamental theorem of calculus, and provide a convenient means for ...

Including:

• Antiderivative - Example
• Antiderivative - Uses and properties
• Antiderivative - Techniques of integration
• Antiderivative - Antiderivatives of non-continuous functions
• Antiderivative - Some examples

Read more here: » Antiderivative: Encyclopedia - Antiderivative

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to comp ...

Including:

• Trigonometric function - List of trigonometric functions
• Trigonometric function - History
• Trigonometric function - Right triangle definitions
• Trigonometric function - Mnemonics
• Trigonometric function - Slope definitions
• Trigonometric function - Unit-circle definitions
• Trigonometric function - Series definitions
• Trigonometric function - Relationship to exponential function
• Trigonometric function - Definitions via differential equations
• Trigonometric function - The significance of radians
• Trigonometric function - Other definitions
• Trigonometric function - Computation
• Trigonometric function - Inverse functions
• Trigonometric function - Identities
• Trigonometric function - Properties and applications
• Trigonometric function - Law of sines
• Trigonometric function - Law of cosines
• Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia - Trigonometric function

Antiderivatives are important because they can be used to compute integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then: Because of this, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f and is written using th ...

See also:

Antiderivative, Antiderivative - Example, Antiderivative - Uses and properties, Antiderivative - Techniques of integration, Antiderivative - Antiderivatives of non-continuous functions, Antiderivative - Some examples

Read more here: » Antiderivative: Encyclopedia II - Antiderivative - Uses and properties

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula s ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - History

Assume (x2 > a2), for (x2 < a2), see next section: Note that , where the positive value of is to be taken. See also:

List of integrals of irrational functions, List of integrals of irrational functions - Integrals Involving, List of integrals of irrational functions - Integrals Involving, List of integrals of irrational functions - Integrals Involving, List of integrals of irrational functions - Integrals Involving, List of integrals of irrational functions - Integrals Involving

Read more here: » List of integrals of irrational functions: Encyclopedia II - List of integrals of irrational functions - Integrals Involving

Like all map projections, attempting to fit a curved surface onto a flat sheet, the shape of the map is a distortion of the true layout of the Earth's surface. The Mercator projection exaggerates the size (and to a lesser extent, the shape) of areas far from the equator. For example, Greenland is presented as being roughly as large as Africa, when in fact Africa's area is approximately 13 times that of Greenland as shown by Tissot's indicatrix. Although the Mercator projection is still in common use for navigation, critics argue that ...

See also:

Mercator projection, Mercator projection - Controversy, Mercator projection - Derivation of the projection

Read more here: » Mercator projection: Encyclopedia II - Mercator projection - Controversy

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula s ...

See also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - History

Please note: Here, and generally in calculus, all angles are measured in radians. (See also below). Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x: These identities are often taken as the definitions of the sine and cosine function. They are often used ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Series definitions

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians. It also provides a single visual picture that encapsulates at once all the important triangles used ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Unit-circle definitions

In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A: We use the following names for the sides of the triangle: The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h. The opposite side is the side opposite to the angle we are interested in, in this case a. The adjacent side is the side that is in contact with the angle we are interested in and the right angl ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Right triangle definitions

Both the sine and cosine functions satisfy the differential equation i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independ ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Definitions via differential equations

Theorem: There exists exactly one pair of real functions s, c with the following properties: For any : ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Other definitions

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results: Trigonometric function - Law of sines. The law of sines for an arbitrary triangle states: It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points ASee also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Properties and applications

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as: For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative in ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Inverse functions

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a sma ...

See also:

Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Computation

A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as: versed sine (versin = 1 − cos) exsecant (exsec = sec − 1). Many more relations between these functions are listed in the article about trigonometric identities. ...

See also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - List of trigonometric functions

Both the sine and cosine functions satisfy the differential equation i.e. each is the additive inverse of its own second derivative. Within the 2-dimensional vector space V consisting of all solutions of this equation, the sine function is the unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and the cosine function is the unique solution satisfying the initial conditions y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly independ ...

See also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Definitions via differential equations

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results: Trigonometric function - Law of sines. The law of sines for an arbitrary triangle states: It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number (sinA)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points ASee also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Properties and applications

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as: For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative in ...

See also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Inverse functions

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a sma ...

See also:

Trigonometric function, Trigonometric function - List of trigonometric functions, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents

Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - Computation