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Encyclopedia > List of integrals of exponential functions

The following is a list of integrals (antiderivative functions) of exponential functions. For a complete list of Integral functions, please see table of integrals and list of integrals. In calculus, the integral of a function is an extension of the concept of a sum. ... 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. ... The exponential function is one of the most important functions in mathematics. ... Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. ... 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...

$int e^{cx};dx = frac{1}{c} e^{cx}$

$int a^{cx};dx = frac{1}{c ln a} a^{cx} qquadmbox{(for } a > 0,mbox{ }a ne 1mbox{)}$

$int xe^{cx}; dx = frac{e^{cx}}{c^2}(cx-1)$

$int x^2 e^{cx};dx = e^{cx}left(frac{x^2}{c}-frac{2x}{c^2}+frac{2}{c^3}right)$

$int x^n e^{cx}; dx = frac{1}{c} x^n e^{cx} - frac{n}{c}int x^{n-1} e^{cx} dx$

$intfrac{e^{cx}; dx}{x} = ln|x| +sum_{i=1}^inftyfrac{(cx)^i}{icdot i!}$

$intfrac{e^{cx}; dx}{x^n} = frac{1}{n-1}left(-frac{e^{cx}}{x^{n-1}}+cintfrac{e^{cx} }{x^{n-1}},dxright) qquadmbox{(for }nneq 1mbox{)}$

$int e^{cx}ln x; dx = frac{1}{c}e^{cx}ln|x|-operatorname{Ei},(cx)$

$int e^{cx}sin bx; dx = frac{e^{cx}}{c^2+b^2}(csin bx - bcos bx)$

$int e^{cx}cos bx; dx = frac{e^{cx}}{c^2+b^2}(ccos bx + bsin bx)$

$int e^{cx}sin^n x; dx = frac{e^{cx}sin^{n-1} x}{c^2+n^2}(csin x-ncos x)+frac{n(n-1)}{c^2+n^2}int e^{cx}sin^{n-2} x;dx$

$int e^{cx}cos^n x; dx = frac{e^{cx}cos^{n-1} x}{c^2+n^2}(ccos x+nsin x)+frac{n(n-1)}{c^2+n^2}int e^{cx}cos^{n-2} x;dx$

$int x e^{c x^2 }; dx= frac{1}{2c} ; e^{c x^2}$

$int {1 over sigmasqrt{2pi} },e^{-{(x-mu )^2 / 2sigma^2}}; dx= frac{1}{2 sigma} (1 + mbox{erf},frac{x-mu}{sigma sqrt{2}})$

$int e^{x^2},dx = e^{x^2}left( sum_{j=0}^{n-1}c_{2j},frac{1}{x^{2j+1}} right )+(2n-1)c_{2n-2} int frac{e^{x^2}}{x^{2n}};dx quad mbox{valid for } n > 0,$

where $c_{2j}=frac{ 1 cdot 3 cdot 5 cdots (2j-1)}{2^{j+1}}=frac{(2j),!}{j!, 2^{2j+1}} .$

$int_{-infty}^{infty} e^{-ax^2},dx=sqrt{pi over a}$ (the Gaussian integral)

$int_{-infty}^{infty} e^{-ax^2} e^{bx},dx=sqrt{frac{pi}{a}}e^{frac{b^2}{4a}}$

$int_{-infty}^{infty} x e^{-a(x-b)^2},dx=b sqrt{pi over a}$

$int_{-infty}^{infty} x^2 e^{-ax^2},dx=frac{1}{2} sqrt{pi over a^3}$

$int_{0}^{infty} x^{2n} e^{-{x^2}/{a^2}},dx=sqrt{pi} {(2n)! over {n!}} {left (frac{a}{2} right)}^{2n + 1}$

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

I 0

is the modified Bessel function of the first kind)

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

$int_{0}^{infty} x^a e^{-bx} dx = frac{a!}{b^{a+1}}$

Categories: Exponentials | Integrals | Mathematics-related lists 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, Bessel functions, first defined by the mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

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