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Encyclopedia > Table of integrals

It has been suggested that this article or section be merged with List of integrals. (Discuss)

Integration is one of the two basic operations in calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. Wikipedia does not have an article with this exact name. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc functions... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... For a non-technical overview of the subject, see Calculus. ... Look up Function in Wiktionary, the free dictionary. ...

This page lists some of the most common antiderivatives; a more complete list can be found in the list of integrals. See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc functions...

We use C for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives. In calculus, the indefinite integral of a given function (i. ...

These formulas only state in another form the assertions in the table of derivatives. The primary operation in differential calculus is finding a derivative. ...

1 Rules for integration of general functions
2 Integrals of simple functions
3 Definite integrals lacking closed-form antiderivatives
- 3.1 The "sophomore's dream"

Rules for integration of general functions

These rules apply only whenever the respective functions are integrable.

$int af(x),dx = aint f(x),dx qquadmbox{(}a neq 0 mbox{, constant)},!$

$int [f(x) + g(x)],dx = int f(x),dx + int g(x),dx$

$int f'(x)g(x),dx = f(x)g(x) - int f(x)g'(x),dx$

$int {f'(x)over f(x)},dx= ln{left|f(x)right|} + C$

$int {f'(x) f(x)},dx= {1 over 2} [ f(x) ]^2 + C$

$int [f(x)]^n f'(x),dx = {[f(x)]^{n+1} over n+1} + C qquadmbox{(for } nneq -1mbox{)},!$

Integrals of simple functions

Rational functions

more integrals: List of integrals of rational functions

$int ,{rm d}x = x + C$

$int x^n,{rm d}x = frac{x^{n+1}}{n+1} + Cqquadmbox{ if }n ne -1$

$int {dx over x} = ln{left|xright|} + C$

$int {dx over {a^2+x^2}} = {1 over a}arctan {x over a} + C$

The following is a list of integrals (antiderivative functions) of rational functions. ...

Irrational functions

more integrals: List of integrals of irrational functions

$int {dx over sqrt{a^2-x^2}} = sin^{-1} {x over a} + C$

$int {-dx over sqrt{a^2-x^2}} = cos^{-1} {x over a} + C$

$int {dx over x sqrt{x^2-a^2}} = {1 over a} sec^{-1} {|x| over a} + C$

The following is a list of integrals (antiderivative functions) of irrational functions. ...

Logarithms

more integrals: List of integrals of logarithmic functions

$int ln {x},dx = x ln {x} - x + C$

$int log_b {x},dx = xlog_b {x} - xlog_b {e} + C$

The following is a list of integrals (antiderivative functions) of logarithmic functions. ...

Exponential functions

more integrals: List of integrals of exponential functions

$int e^x,dx = e^x + C$

$int a^x,dx = frac{a^x}{ln{a}} + C$

The following is a list of integrals (antiderivative functions) of exponential functions. ...

Trigonometric functions

more integrals: List of integrals of trigonometric functions and List of integrals of arc functions

$int sin{x}, dx = -cos{x} + C$

$int cos{x}, dx = sin{x} + C$

$int tan{x} , dx = -ln{left| cos {x} right|} + C$

$int cot{x} , dx = ln{left| sin{x} right|} + C$

$int sec{x} , dx = ln{left| sec{x} + tan{x}right|} + C$

$int csc{x} , dx = ln{left| csc{x} - cot{x}right|} + C$

$int sec^2 x , dx = tan x + C$

$int csc^2 x , dx = -cot x + C$

$int sec{x} , tan{x} , dx = sec{x} + C$

$int csc{x} , cot{x} , dx = - csc{x} + C$

$int sin^2 x , dx = frac{1}{2}(x - sin x cos x) + C$

$int cos^2 x , dx = frac{1}{2}(x + sin x cos x) + C$

$int sec^3 x , dx = frac{1}{2}sec x tan x + frac{1}{2}ln|sec x + tan x| + C$

(see integral of secant cubed)

$int sin^n x , dx = - frac{sin^{n-1} {x} cos {x}}{n} + frac{n-1}{n} int sin^{n-2}{x} , dx$

$int cos^n x , dx = frac{cos^{n-1} {x} sin {x}}{n} + frac{n-1}{n} int cos^{n-2}{x} , dx$

$int arctan{x} , dx = x , arctan{x} - frac{1}{2} ln{left| 1 + x^2right|} + C$

The following is a list of integrals (antiderivative functions) of trigonometric functions. ... The following is a list of integrals (antiderivative formulas) for integrands that contain inverse trigonometric functions (also know as â€œarc functionsâ€). For a complete list of integral formulas, see the Table of Integrals and the List of Integrals. ... One of the more challenging indefinite integrals of elementary calculus is This antiderivative may be found by integration by parts, as follows: where Then Next we add to both sides of the equality just derived: Then divide both sides by 2. ...

Hyperbolic functions

more integrals: List of integrals of hyperbolic functions

$int sinh x , dx = cosh x + C$

$int cosh x , dx = sinh x + C$

$int tanh x , dx = ln| cosh x | + C$

$int mbox{csch},x , dx = lnleft| tanh {x over2}right| + C$

$int mbox{sech},x , dx = arctan(sinh x) + C$

$int coth x , dx = ln| sinh x | + C$

$int mbox{sech}^2 x, dx = tanh x + C$

The following is a list of integrals (antiderivative functions) of hyperbolic functions. ...

Inverse hyperbolic functions

$int operatorname{arcsinh} x , dx = x operatorname{arcsinh} x - sqrt{x^2+1} + C$

$int operatorname{arccosh} x , dx = x operatorname{arccosh} x - sqrt{x^2-1} + C$

$int operatorname{arctanh} x , dx = x operatorname{arctanh} x + frac{1}{2}log{(1-x^2)} + C$

$int operatorname{arccsch},x , dx = x operatorname{arccsch} x+ log{left[xleft(sqrt{1+frac{1}{x^2}} + 1right)right]} + C$

$int operatorname{arcsech},x , dx = x operatorname{arcsech} x- arctan{left(frac{x}{x-1}sqrt{frac{1-x}{1+x}}right)} + C$

$int operatorname{arccoth},x , dx = x operatorname{arccoth} x+ frac{1}{2}log{(x^2-1)} + C$

Definite integrals lacking closed-form antiderivatives

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.

$int_0^infty{sqrt{x},e^{-x},dx} = frac{1}{2}sqrt pi$ (see also Gamma function)

$int_0^infty{e^{-x^2},dx} = frac{1}{2}sqrt pi$ (the Gaussian integral)

$int_0^infty{frac{x}{e^x-1},dx} = frac{pi^2}{6}$ (see also Bernoulli number)

$int_0^infty{frac{x^3}{e^x-1},dx} = frac{pi^4}{15}$

$int_0^inftyfrac{sin(x)}{x},dx=frac{pi}{2}$

$int_0^frac{pi}{2}sin^n{x},dx=int_0^frac{pi}{2}cos^n{x},dx=frac{1 cdot 3 cdot 5 cdot cdots cdot (n-1)}{2 cdot 4 cdot 6 cdot cdots cdot n}frac{pi}{2}$ (if n is an even integer and $scriptstyle{n ge 2}$ )

$int_0^frac{pi}{2}sin^n{x},dx=int_0^frac{pi}{2}cos^n{x},dx=frac{2 cdot 4 cdot 6 cdot cdots cdot (n-1)}{3 cdot 5 cdot 7 cdot cdots cdot n}$ (if $scriptstyle{n}$ is an odd integer and $scriptstyle{n ge 3}$ )

$int_0^inftyfrac{sin^2{x}}{x^2},dx=frac{pi}{2}$

$int_0^infty x^{z-1},e^{-x},dx = Gamma(z)$ (where

Γ(z)

is the Gamma function)

$int_{-infty}^infty e^{-(ax^2+bx+c)},dx=sqrt{frac{pi}{a}}expleft[frac{b^2-4ac}{4a}right]$ (where

exp[u]

is the exponential function

e u

$int_{0}^{2 pi} e^{x cos theta} d theta = 2 pi I_{0}(x)$ (where

I 0 (x)

is the modified Bessel function of the first kind)

$int_{0}^{2 pi} e^{x cos theta + y sin theta} d theta = 2 pi I_{0} left(sqrt{x^2 + y^2}right)$

$int_{-infty}^{infty}{(1 + x^2/nu)^{-(nu + 1)/2}dx} = frac { sqrt{nu pi} Gamma(nu/2)} {Gamma((nu + 1)/2))},$ ( $nu > 0,$ , this is related to the probability density function of the Student's t-distribution)

The method of exhaustion provides a formula for the general case when no antiderivative exists: The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ... In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred to... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. ...

$int_a^b{f(x),dx} = (b - a) sumlimits_{n = 1}^infty {sumlimits_{m = 1}^{2^n - 1} {left( { - 1} right)^{m + 1} } } 2^{ - n} f(a + mleft( {b - a} right)2^{-n} )$

The "sophomore's dream"

$begin{align} int_0^1 x^{-x},dx &= sum_{n=1}^infty n^{-n} &&(= 1.291285997dots) int_0^1 x^x ,dx &= sum_{n=1}^infty -(-1)^nn^{-n} &&(= 0.783430510712dots) end{align}$

(attributed to Johann Bernoulli; see sophomore's dream). Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... In mathematics, sophomores dream is a name occasionally used for the identities discovered in 1697 by Johann Bernoulli. ...

Categories: Articles to be merged since January 2007 | Integrals | Mathematics-related lists

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