The following is a list of
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s of
exponential functions. For a complete list of integral functions, please see the
list of integrals
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, ...
.
Indefinite integral
Indefinite integrals are
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
functions. A constant (the
constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.
Integrals of polynomials
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Integrals involving only exponential functions
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Integrals involving the error function
In the following formulas, is the
error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as:
\operatorname z = \frac\int_0^z e^\,\mathrm dt.
The integral here is a complex Contour integrat ...
and is the
exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of&nb ...
.
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Other integrals
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where
(Note that the value of the expression is ''independent'' of the value of , which is why it does not appear in the integral.)
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where
and is the
upper incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, whic ...
.
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when
,
, and
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when
,
, and
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Definite integrals
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The last expression is the
logarithmic mean
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient.
This calculation is applicable in engineering problems involving heat and mass t ...
.
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(the
Gaussian integral
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
\int_^\infty e^\,dx = \s ...
)
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(see
Integral of a Gaussian function)
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(the operator is the
Double factorial
In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is,
n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
Restated ...
)
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