List of integrals of exponential functions

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

: $\int xe^\,dx = e^\left(\frac\right) \text c \neq 0;$

: $\int x^2 e^\,dx = e^\left(\frac-\frac+\frac\right)$

: $\begin \int x^n e^\,dx &= \frac x^n e^ - \frac\int x^ e^ \,dx \\ &= \left( \frac \right)^n \frac \\ &= e^\sum_^n (-1)^i\fracx^ \\ &= e^\sum_^n (-1)^\fracx^i \end$

: $\int\frac\,dx = \ln, x, +\sum_^\infty\frac$

: $\int\frac\,dx = \frac\left(-\frac+c\int\frac\,dx\right) \qquad\textn\neq 1\text$

Integrals involving only exponential functions

: $\int f'(x)e^\,dx = e^$

: $\int e^\,dx = \frac e^$

: $\int a^\,dx = \frac a^\qquad\texta > 0,\ a \ne 1$

Integrals involving the error function

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

: $\int e^\ln x\,dx = \frac\left(e^\ln, x, -\operatorname(cx)\right)$

:$\int x e^\,dx= \frac e^$

:$\int e^\,dx= \sqrt \operatorname(\sqrt x)$

:$\int xe^\,dx=-\frace^$

: $\int\frac\,dx = -\frac - \sqrt \operatorname (x)$

:$\int \,dx= \frac\operatorname\left(\frac\right)$

Other integrals

:$\int e^\,dx = e^\left( \sum_^c_\frac \right )+(2n-1)c_ \int \frac\,dx \quad \text n > 0,$
::where $c_=\frac=\frac \ .$
::(Note that the value of the expression is ''independent'' of the value of , which is why it does not appear in the integral.)

:$$
:: where $a_=\begin1 &\text n = 0, \\ \\ \dfrac &\text m=1, \\ \\ \dfrac\sum_^ja_a_ &\text \end$
:: and is the upper incomplete gamma function.

:$\int \frac \,dx = \frac - \frac \ln\left(a e^ + b \right)$ when $b \neq 0$, $\lambda \neq 0$, and $ae^ + b > 0.$

:\int \frac \,dx = \frac \left e^ + b - b \ln\left(a e^ + b \right) \right when $a \neq 0$, $\lambda \neq 0$, and $ae^ + b > 0.$

:$\int \frac\,dx=\frac+x.$

:$\int = e^f\left( x \right) + C$
the below formulae was proved by Toyesh Prakash Sharma.
:$\int = e^\sum_^ + C$(if $n$ is a positive integer)
:$\int = - e^\sum_^\frac + C$(if $n$ is a positive integer)

Definite integrals

: $\begin \int_0^1 e^\,dx &= \int_0^1 \left(\frac\right)^\cdot b\,dx \\ &= \int_0^1 a^\cdot b^\,dx \\ &= \frac \qquad\text a > 0,\ b > 0,\ a \neq b \end$
The last expression is the logarithmic mean.

:$\int_0^ e^\,dx=\frac \quad (\operatorname(a)>0)$

:$\int_0^ e^\,dx=\frac \sqrt \quad (a>0)$ (the Gaussian integral)

:$\int_^ e^\,dx=\sqrt \quad (a>0)$
:$\int_^ e^ e^\,dx=\sqrte^ \quad (a,b>0)$
:
:$\int_^ e^\,dx= \sqrte^ \quad(a > 0)$
$\int_^ e^\,dx= \sqrte^ \quad(a > 0)$
:$\int_^ e^ e^\,dx=\sqrte^ \quad (a>0)$ (see Integral of a Gaussian function)

:$\int_^ x e^\,dx= b \sqrt \quad (\operatorname(a)>0)$

:$\int_^ x e^\,dx= \frac e^ \quad (\operatorname(a)>0)$

:$\int_^ x^2 e^\,dx=\frac \sqrt \quad (a>0)$

:$\int_^ x^2 e^\,dx=\frac e^ \quad (\operatorname(a)>0)$

:$\int_^ x^3 e^\,dx=\frac e^ \quad (\operatorname(a)>0)$

:$\int_0^ x^ e^\,dx = \begin \dfrac & (n>-1,\ a>0) \\ \\ \dfrac\sqrt & (n=2k,\ k \text,\ a>0) \ \text \\ \\ \dfrac & (n=2k+1,\ k \text,\ a>0) \end$
(the operator $!!$ is the Double factorial)

:$\int_0^ x^n e^\,dx = \begin \dfrac & (n>-1,\ \operatorname(a)>0) \\ \\ \dfrac & (n=0,1,2,\ldots,\ \operatorname(a)>0) \end$

:\int_0^ x^n e^\,dx =
\frac\left 1-e^\sum_^ \frac
\right/math>

:\int_0^ x^n e^\,dx =
\frac\left 1-e^\sum_^ \frac
\right/math>

:$\int_0^\infty e^ dx = \frac\ a^\Gamma\left(\frac\right)$

:$\int_0^\infty x^n e^ dx = \frac\ a^\Gamma\left(\frac\right)$

:$\int_0^ e^\sin bx\,dx = \frac \quad (a>0)$

:$\int_0^ e^\cos bx\,dx = \frac \quad (a>0)$

:$\int_0^ xe^\sin bx\,dx = \frac \quad (a>0)$

:$\int_0^ xe^\cos bx\,dx = \frac \quad (a>0)$
:$\int_0^ \frac\,dx=\arctan \frac$
:$\int_0^ \frac\,dx=\ln \frac$
:$\int_0^ \frac \sin px \, dx=\arctan \frac - \arctan \frac$
:$\int_0^ \frac \cos px \, dx=\frac \ln \frac$
:$\int_0^ \frac\,dx=\arccot a - \frac\ln (a^2+1)$
:$\int_0^ e^ d \theta = 2 \pi I_0(x)$ ( is the modified Bessel function of the first kind)
:$\int_0^ e^ d \theta = 2 \pi I_0 \left( \sqrt \right)$
:$\int_0^\infty\frac \,dx = \operatorname_(z)\Gamma(s),$

where $\operatorname_(z)$ is the Polylogarithm.

:$\int_0^\infty\frac \,dx = \frac \coth \frac - \frac$
:$\int_0^\infty e^ \ln x\, dx = - \gamma,$

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

Finally, a well known result,

:$\int_0^ e^ d\phi = 2 \pi \delta_$ (For integer m, n)

where $\delta_$ is the Kronecker delta.

See also

* Gradshteyn and Ryzhik

Further reading

*
*
* Toyesh Prakash Sharma, https://www.isroset.org/pdf_paper_view.php?paper_id=2214&7-ISROSET-IJSRMSS-05130.pdf

External links

Wolfram Mathematica Online Integrator
*

{{DEFAULTSORT:Integrals Of Exponential Functions
Exponentials
Exponential functions