List of integrals of exponential functions

Template:Short description The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals.

Indefinite integral

Indefinite integrals are antiderivative functions. A constant (the constant of integration) 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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$\int x e^{c x} d x = e^{c x} (\frac{c x - 1}{c^{2}}) for c \neq 0;$
$\int x^{2} e^{c x} d x = e^{c x} (\frac{x^{2}}{c} - \frac{2 x}{c^{2}} + \frac{2}{c^{3}})$
$\begin{matrix} \int x^{n} e^{c x} d x & = \frac{1}{c} x^{n} e^{c x} - \frac{n}{c} \int x^{n - 1} e^{c x} d x \\ = {(\frac{\partial}{\partial c})}^{n} \frac{e^{c x}}{c} \\ = e^{c x} \sum_{i = 0}^{n} (- 1)^{i} \frac{n!}{(n - i)! c^{i + 1}} x^{n - i} \\ = e^{c x} \sum_{i = 0}^{n} (- 1)^{n - i} \frac{n!}{i! c^{n - i + 1}} x^{i} \end{matrix}$
$\int \frac{e^{c x}}{x} d x = \ln | x | + \sum_{n = 1}^{\infty} \frac{(c x)^{n}}{n \cdot n!}$
$\int \frac{e^{c x}}{x^{n}} d x = \frac{1}{n - 1} (- \frac{e^{c x}}{x^{n - 1}} + c \int \frac{e^{c x}}{x^{n - 1}} d x) (for n \neq 1)$

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Integrals involving only exponential functions

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$\int f^{'} (x) e^{f (x)} d x = e^{f (x)}$
$\int e^{c x} d x = \frac{1}{c} e^{c x}$
$\int a^{x} d x = \frac{a^{x}}{\ln a} for a > 0, a \neq 1$

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Integrals involving the error function

In the following formulas, Template:Math is the error function and Template:Math is the exponential integral.

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$\int e^{c x} \ln x d x = \frac{1}{c} (e^{c x} \ln | x | - Ei (c x))$
$\int x e^{c x^{2}} d x = \frac{1}{2 c} e^{c x^{2}}$
$\int e^{- c x^{2}} d x = \sqrt{\frac{π}{4 c}} \erf (\sqrt{c} x)$
$\int x e^{- c x^{2}} d x = - \frac{1}{2 c} e^{- c x^{2}}$
$\int \frac{e^{- x^{2}}}{x^{2}} d x = - \frac{e^{- x^{2}}}{x} - \sqrt{π} \erf (x)$
$\int \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} {(\frac{x - μ}{σ})}^{2}} d x = \frac{1}{2} \erf (\frac{x - μ}{σ \sqrt{2}})$

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Other integrals

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$\int e^{x^{2}} d x = e^{x^{2}} (\sum_{j = 0}^{n - 1} c_{2 j} \frac{1}{x^{2 j + 1}}) + (2 n - 1) c_{2 n - 2} \int \frac{e^{x^{2}}}{x^{2 n}} d x valid for any n > 0,$
where $c_{2 j} = \frac{1 \cdot 3 \cdot 5 \dots (2 j - 1)}{2^{j + 1}} = \frac{(2 j)!}{j! 2^{2 j + 1}} .$

(Note that the value of the expression is independent of the value of Template:Mvar, which is why it does not appear in the integral.)

$\int \underset{m}{\underset{⏟}{x^{x^{\cdot^{\cdot^{x}}}}}} d x = \sum_{n = 0}^{m} \frac{(- 1)^{n} (n + 1)^{n - 1}}{n!} Γ (n + 1, - \ln x) + \sum_{n = m + 1}^{\infty} (- 1)^{n} a_{m n} Γ (n + 1, - \ln x) (for x > 0)$

where $a_{m n} = {\begin{matrix} 1 & if n = 0, \\ \frac{1}{n!} & if m = 1, \\ \frac{1}{n} \sum_{j = 1}^{n} j a_{m, n - j} a_{m - 1, j - 1} & otherwise \end{matrix}$

and Template:Math is the upper incomplete gamma function.

$\int \frac{1}{a e^{λ x} + b} d x = \frac{x}{b} - \frac{1}{b λ} \ln (a e^{λ x} + b)$ when $b \neq 0$ , $λ \neq 0$ , and $a e^{λ x} + b > 0 .$
$\int \frac{e^{2 λ x}}{a e^{λ x} + b} d x = \frac{1}{a^{2} λ} [a e^{λ x} + b - b \ln (a e^{λ x} + b)]$ when $a \neq 0$ , $λ \neq 0$ , and $a e^{λ x} + b > 0 .$
$\int \frac{a e^{c x} - 1}{b e^{c x} - 1} d x = \frac{(a - b) \log (1 - b e^{c x})}{b c} + x .$
$\int e^{x} (f (x) + f^{'} (x)) dx = e^{x} f (x) + C$

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$\int e^{x} (f (x) - {(- 1)}^{n} \frac{d^{n} f (x)}{d x^{n}}) d x = e^{x} \sum_{k = 1}^{n} {(- 1)}^{k - 1} \frac{d^{k - 1} f (x)}{d x^{k - 1}} + C$
$\int e^{- x} (f (x) - \frac{d^{n} f (x)}{d x^{n}}) d x = - e^{- x} \sum_{k = 1}^{n} \frac{d^{k - 1} f (x)}{d x^{k - 1}} + C$

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$\int e^{a x} ({(a)}^{n} f (x) - {(- 1)}^{n} \frac{d^{n} f (x)}{d x^{n}}) d x = e^{a x} \sum_{k = 1}^{n} {(a)}^{n - k} {(- 1)}^{k - 1} \frac{d^{k - 1} f (x)}{d x^{k - 1}} + C$

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Definite integrals

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$\begin{matrix} \int_{0}^{1} e^{x \cdot \ln a + (1 - x) \cdot \ln b} d x & = \int_{0}^{1} {(\frac{a}{b})}^{x} \cdot b d x \\ = \int_{0}^{1} a^{x} \cdot b^{1 - x} d x \\ = \frac{a - b}{\ln a - \ln b} for a > 0, b > 0, a \neq b \end{matrix}$

The last expression is the logarithmic mean.

$\int_{0}^{\infty} e^{- a x} d x = \frac{1}{a} (Re (a) > 0)$
$\int_{0}^{\infty} e^{- a x^{2}} d x = \frac{1}{2} \sqrt{\frac{π}{a}} (a > 0)$ (the Gaussian integral)
$\int_{- \infty}^{\infty} e^{- a x^{2}} d x = \sqrt{\frac{π}{a}} (a > 0)$
$\int_{- \infty}^{\infty} e^{- a x^{2}} e^{- \frac{b}{x^{2}}} d x = \sqrt{\frac{π}{a}} e^{- 2 \sqrt{a b}} (a, b > 0)$
$\int_{- \infty}^{\infty} e^{- (a x^{2} + b x)} d x = \sqrt{\frac{π}{a}} e^{\frac{b^{2}}{4 a}} (a > 0)$
$\int_{- \infty}^{\infty} e^{- (a x^{2} + b x + c)} d x = \sqrt{\frac{π}{a}} e^{\frac{b^{2}}{4 a} - c} (a > 0)$
$\int_{- \infty}^{\infty} e^{- a x^{2}} e^{- 2 b x} d x = \sqrt{\frac{π}{a}} e^{\frac{b^{2}}{a}} (a > 0)$ (see Integral of a Gaussian function)
$\int_{- \infty}^{\infty} x e^{- a (x - b)^{2}} d x = b \sqrt{\frac{π}{a}} (Re (a) > 0)$
$\int_{- \infty}^{\infty} x e^{- a x^{2} + b x} d x = \frac{\sqrt{π} b}{2 a^{3 / 2}} e^{\frac{b^{2}}{4 a}} (Re (a) > 0)$
$\int_{- \infty}^{\infty} x^{2} e^{- a x^{2}} d x = \frac{1}{2} \sqrt{\frac{π}{a^{3}}} (a > 0)$
$\int_{- \infty}^{\infty} x^{2} e^{- (a x^{2} + b x)} d x = \frac{\sqrt{π} (2 a + b^{2})}{4 a^{5 / 2}} e^{\frac{b^{2}}{4 a}} (Re (a) > 0)$
$\int_{- \infty}^{\infty} x^{3} e^{- (a x^{2} + b x)} d x = \frac{\sqrt{π} (6 a + b^{2}) b}{8 a^{7 / 2}} e^{\frac{b^{2}}{4 a}} (Re (a) > 0)$
$\int_{0}^{\infty} x^{n} e^{- a x^{2}} d x = {\begin{matrix} \frac{Γ (\frac{n + 1}{2})}{2 (a^{\frac{n + 1}{2}})} & (n > - 1, a > 0) \\ \frac{(2 k - 1)!!}{2^{k + 1} a^{k}} \sqrt{\frac{π}{a}} & (n = 2 k, k integer, a > 0) \\ \frac{k!}{2 (a^{k + 1})} & (n = 2 k + 1, k integer, a > 0) \end{matrix}$

(the operator $!!$ is the Double factorial)

$\int_{0}^{\infty} x^{n} e^{- a x} d x = {\begin{matrix} \frac{Γ (n + 1)}{a^{n + 1}} & (n > - 1, Re (a) > 0) \\ \frac{n!}{a^{n + 1}} & (n = 0, 1, 2, \dots, Re (a) > 0) \end{matrix}$
$\int_{0}^{1} x^{n} e^{- a x} d x = \frac{n!}{a^{n + 1}} [1 - e^{- a} \sum_{i = 0}^{n} \frac{a^{i}}{i!}]$
$\int_{0}^{b} x^{n} e^{- a x} d x = \frac{n!}{a^{n + 1}} [1 - e^{- a b} \sum_{i = 0}^{n} \frac{(a b)^{i}}{i!}]$
$\int_{0}^{\infty} e^{- a x^{b}} d x = \frac{1}{b} a^{- \frac{1}{b}} Γ (\frac{1}{b})$
$\int_{0}^{\infty} x^{n} e^{- a x^{b}} d x = \frac{1}{b} a^{- \frac{n + 1}{b}} Γ (\frac{n + 1}{b})$
$\int_{0}^{\infty} e^{- a x} \sin b x d x = \frac{b}{a^{2} + b^{2}} (a > 0)$
$\int_{0}^{\infty} e^{- a x} \cos b x d x = \frac{a}{a^{2} + b^{2}} (a > 0)$
$\int_{0}^{\infty} x e^{- a x} \sin b x d x = \frac{2 a b}{(a^{2} + b^{2})^{2}} (a > 0)$
$\int_{0}^{\infty} x e^{- a x} \cos b x d x = \frac{a^{2} - b^{2}}{(a^{2} + b^{2})^{2}} (a > 0)$
$\int_{0}^{\infty} \frac{e^{- a x} \sin b x}{x} d x = \arctan \frac{b}{a}$
$\int_{0}^{\infty} \frac{e^{- a x} - e^{- b x}}{x} d x = \ln \frac{b}{a}$
$\int_{0}^{\infty} \frac{e^{- a x} - e^{- b x}}{x} \sin p x d x = \arctan \frac{b}{p} - \arctan \frac{a}{p}$
$\int_{0}^{\infty} \frac{e^{- a x} - e^{- b x}}{x} \cos p x d x = \frac{1}{2} \ln \frac{b^{2} + p^{2}}{a^{2} + p^{2}}$
$\int_{0}^{\infty} \frac{e^{- a x} (1 - \cos x)}{x^{2}} d x = arccot a - \frac{a}{2} \ln (\frac{1}{a^{2}} + 1)$
$\int_{- \infty}^{\infty} e^{a x^{4} + b x^{3} + c x^{2} + d x + f} d x = e^{f} \sum_{n, m, p = 0}^{\infty} \frac{b^{4 n}}{(4 n)!} \frac{c^{2 m}}{(2 m)!} \frac{d^{4 p}}{(4 p)!} \frac{Γ (3 n + m + p + \frac{1}{4})}{a^{3 n + m + p + \frac{1}{4}}}$ (appears in several models of extended superstring theory in higher dimensions)
$\int_{0}^{2 π} e^{x \cos θ} d θ = 2 π I_{0} (x)$ (Template:Math is the modified Bessel function of the first kind)
$\int_{0}^{2 π} e^{x \cos θ + y \sin θ} d θ = 2 π I_{0} (\sqrt{x^{2} + y^{2}})$
$\int_{0}^{\infty} \frac{x^{s - 1}}{e^{x} / z - 1} d x = {Li}_{s} (z) Γ (s),$

where ${Li}_{s} (z)$ is the Polylogarithm.

$\int_{0}^{\infty} \frac{\sin m x}{e^{2 π x} - 1} d x = \frac{1}{4} \coth \frac{m}{2} - \frac{1}{2 m}$
$\int_{0}^{\infty} e^{- x} \ln x d x = - γ,$

where $γ$ is the Euler–Mascheroni constant which equals the value of a number of definite integrals.

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Finally, a well known result, $\int_{0}^{2 π} e^{i (m - n) ϕ} d ϕ = 2 π δ_{m, n} for m, n \in ℤ$ where $δ_{m, n}$ is the Kronecker delta.

References

Template:Reflist Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.[1]

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Template:Lists of integrals

List of integrals of exponential functions

Contents

Indefinite integral

Integrals of polynomials

Integrals involving only exponential functions

Integrals involving the error function

Other integrals

Definite integrals

See also

References

Further reading

External links

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List of integrals of exponential functions

Indefinite integral

Integrals of polynomials

Integrals involving only exponential functions

Integrals involving the error function

Other integrals

Definite integrals

See also

References

Further reading

External links

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