List of integrals of exponential functions explained

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 ^{cx} dx = e ^{cx} \left( (cx -1)/(c ^{2}) \right) for c ≠ 0 ; \int x ^{2} e ^{cx} dx = e ^{cx} \left( (x ^{2})/(- c) (2 x)/(+ c ^{2}) (2)/(c ^{3}) \right) \begin{align} \int x ^{n} e ^{cx} dx & = (1)/(c) x ^{n} e ^{cx} - (n)/(c) \int x ^{n-1} e ^{cx} dx \\ & = \left( (\partial)/(\partial c) \right) ^{n} (e ^{cx})/(c) \\ & = e ^{cx} \sum ^{n} _{i =0} ( -1 ) ^{i (n!)/((n - i)!c ^{i+1})} x ^{n-i} \\ & = e ^{cx} \sum ^{n} _{i =0} ( -1 ) ^{n-i} (n!)/(i!c ^{n-i+1}) x ^{i} \end{align} \int (e ^{cx})/(x) dx = ln |x| + \sum ^{inf ty ((cx) ^{n})/(n ⋅ n!)} _{n =1} \int (e ^{cx})/(x ^{n}) dx = (1)/(\left( - n -1) (e ^{cx})/(+ c\int x ^{n-1}) (e ^{cx})/(x ^{n-1}) dx\right) (for n ≠ 1 )

Integrals involving only exponential functions

\int f'(x)e ^{f(x)} dx = e ^{f(x)} \int e ^{cx} dx = (1)/(c) e ^{cx} \int a ^{x} dx = (a ^{x})/(ln a) for a > 0 , a \ne 1

Integrals involving the error function

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

\int e ^{cx} ln xdx = (1)/(c) \left(e ^{cx} ln |x| - \operatorname{Ei}(cx)\right) \int x e ^{c x ^{2}} dx = (1)/(2 c) e ^{c x ^{2}} \int e ^{- c x ^{2}} dx = \sqrt{ (\pi)/(4 c) } \operatorname(\sqrt x)\int xe ^{- c x ^{2}} dx =- (1)/(2 c) e ^{- cx ^{2}} \int (e ^{- x ^{2}})/(x ^{2}) dx =- (e ^{- x ^{2}})/(x) - \sqrt{\pi} \operatorname{erf} (x) \int { (1)/(\sigma\sqrt{ 2 \pi) } e^}\,dx= \frac\operatorname\left(\frac\right)

Other integrals

\int e ^{x ^{2}} dx = e ^{x ^{2}} \left( \sum ^{n -1} _{j =0} c _{2j} (1)/(x ^{2j+1}) \right ) + ( 2 n -1 )c _{2n-2} \int (e ^{x ^{2}})/(x ^{2n}) dx validforany n > 0 ,

where

c _{2j} = (1 ⋅ 3 ⋅ 5 … ( 2 j -1 ))/(= 2 ^{j+1}) (( 2 j)!)/(j! 2 ^{2j+1}) .

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

{\int \underbrace{x ^{x ^{⋅ ^{⋅ ^{x}}}} }_mdx= \sum_^m\frac\Gamma(n+1,- \ln x) + \sum_^\infty(-1)^na_\Gamma(n+1,-\ln x) \qquad\textx> 0\text}

where

a _{mn} = \begin{cases} 1 & if n = 0 , \ \ \dfrac{ 1 }{n!} & if m =1 , \ \ \dfrac{ 1 }{n}\sum ^{n} _{j =1} ja _{m,n-j} a _{m-1,j-1} & otherwise \end{cases}

and is the upper incomplete gamma function.

\int (1)/(ae ^{λ} + b) dx = (x)/(b) - (1)/(b λ) ln \left(a e ^{λ} + b \right) when b ≠ 0 , λ ≠ 0 , and ae ^{λ} + b > 0 . \int (e ^{2λ})/(ae ^{λ} + b) dx = (1)/(a ^{2} λ) \left[a e ^{λ} + b - b ln \left(a e ^{λ} + b \right) \right] when a ≠ 0 , λ ≠ 0 , and ae ^{λ} + b > 0 . \int (ae ^{cx} -1)/(dx = be ^{cx} -1) ((a - b) log ( 1- be ^{cx} ))/(bc) + x. \int{e ^{x} \left( f\left( x \right) + f'\left( x \right) \right) dx } = e^f\left(x \right) + C\int {e ^{x} \left( f\left( x \right) - \left( - 1 \right) ^{n} (d ^{n} f\left( x \right))/(dx ^{n}) \right)dx} = e ^{x} \sum ^{n} _{k =1} {\left( - 1 \right) ^{k} (d ^{k} f\left( x \right))/(dx ^{k}) } + C\int {e ^{-} \left( f\left( x \right) - (d ^{n} f\left( x \right))/(dx ^{n}) \right) dx} = - e ^{-} \sum ^{n} _{k =1} (d ^{k} f\left( x \right))/(dx ^{k}) + C \int {e ^{ax} \left( \left( a\right) ^{n} f\left( x \right) - \left( - 1 \right) ^{n} (d ^{n} f\left( x \right))/(dx ^{n}) \right)dx} = e ^{ax} \sum ^{n} _{k =1} {\left(a\right) ^{n-k} \left( - 1 \right) ^{k} (d ^{k} f\left( x \right))/(dx ^{k}) } + C

Definite integrals

\begin{align} \int ^{1} _{0} e ^{x ⋅} dx & = \int ^{1} _{0} \left( (a)/(b) \right) ^{x} ⋅ bdx \\ & = \int ^{1} _{0} a ^{x} ⋅ b ^{1-x} dx \\ & = (a - b)/(ln a -ln b) for a > 0 , b > 0 , a ≠ b \end{align}

The last expression is the logarithmic mean.

\int ^{inf ty} _{0} e ^{-ax} dx = (1)/(a) (\operatorname{Re}(a)> 0 ) \int ^{inf ty} _{0} e ^{- ax ^{2}} dx = (1)/(2) \sqrt{\pi \over a} (a> 0 ) (the Gaussian integral)\int ^{inf ty} _{-inf ty} e ^{- ax ^{2}} dx = \sqrt{\pi \over a} (a> 0 ) \int ^{inf ty} _{-inf ty} e ^{- ax ^{2}} e ^{- (b)/(x ^{2})} dx = \sqrt{ (\pi)/(a) }e^ \quad (a,b>0) \int ^{inf ty} _{-inf ty} e ^{- (ax ^{2} + bx)} dx = \sqrt{\pi \over a}e ^{\tfrac{b ^{2}} { 4 a}} (a > 0 ) \int ^{inf ty} _{-inf ty} e ^{- (ax ^{2} + bx + c)} dx = \sqrt{\pi \over a}e ^{\tfrac{b ^{2}} { 4 a} - c} (a > 0 ) \int ^{inf ty} _{-inf ty} e ^{- ax ^{2}} e ^{-2bx} dx = \sqrt{ (\pi)/(a) }e^ \quad (a>0) (see Integral of a Gaussian function)\int ^{inf ty} _{-inf ty} x e ^{- a(x - b) ^{2}} dx = b \sqrt{ (\pi)/(a) } \quad (\operatorname(a)>0)\int ^{inf ty} _{-inf ty} x e ^{- ax ^{2+bx}} dx = (\sqrt{\pi)/(b) }{ 2 a ^{3/2} } e^ \quad (\operatorname(a)>0)\int ^{inf ty} _{-inf ty} x ^{2} e ^{- ax ^{2}} dx = (1)/(2) \sqrt{\pi \over a ^{3}} (a> 0 ) \int ^{inf ty} _{-inf ty} x ^{2} e ^{- (ax ^{2+bx)}} dx = (\sqrt{\pi)/(( 2 a + b) ^{2)}{4a} ^{5/2} } e^ \quad (\operatorname(a)>0)\int ^{inf ty} _{-inf ty} x ^{3} e ^{- (ax ^{2+bx)}} dx = (\sqrt{\pi)/(( 6 a + b) ^{2)b}{8a} ^{7/2} } e^ \quad (\operatorname(a)>0)\int ^{inf ty} _{0} x ^{n} e ^{- ax ^{2}} dx = \begin{cases} \dfrac{\Gamma \left( (n +1)/(2) \right)}{ 2 \left(a ^{(n +1)/(2)} \right) } & (n> -1 , a> 0 ) \\ \dfrac{( 2 k -1 )!!}{ 2 ^{k+1} a ^{k}\sqrt{\dfrac{\pi}{a}}} & (n =2 k, k integer , a> 0 ) \\ \dfrac{k!}{ 2 (a ^{k+1} )} & (n =2 k +1 , k integer , a> 0 ) \end{cases}

(the operator

!! is the Double factorial)

\int ^{inf ty} _{0} x ^{n} e ^{-ax} dx = \begin{cases} \dfrac{\Gamma(n +1 )}{a ^{n+1} } & (n>-1,\ \operatorname(a)>0) \\ \\ \dfrac & (n=0,1,2,\ldots,\ \operatorname(a)>0) \end\int ^{1} _{0} x ^{n} e ^{-ax} dx = (n!)/(a ^{n+1}) \left[ 1- e ^{-a} \sum ^{n} _{i =0} (a ^{i})/(i!) \right] \int ^{b} _{0} x ^{n} e ^{-ax} dx = (n!)/(a ^{n+1}) \left[ 1- e ^{-ab} \sum ^{n} _{i =0} ((ab) ^{i})/(i!) \right] \int ^{inf ty} _{0} e ^{- ax ^{b}} dx = (1)/(b) a ^{- (1)/(b)} \Gamma\left( (1)/(b) \right) \int ^{inf ty} _{0} x ^{n} e ^{- ax ^{b}} dx = (1)/(b) a ^{- (n +1)/(b)} \Gamma\left( (n +1)/(b) \right) \int ^{inf ty} _{0} e ^{-ax} \sin bxdx = (b)/(a ^{2+b} ^{2}) (a> 0 ) \int ^{inf ty} _{0} e ^{-ax} \cos bxdx = (a)/(a ^{2+b} ^{2}) (a> 0 ) \int ^{inf ty} _{0} xe ^{-ax} \sin bxdx = (2 ab)/((a ^{2+b} ^{2)} ^{2}) (a> 0 ) \int ^{inf ty} _{0} xe ^{-ax} \cos bxdx = (a ^{2-b} ^{2})/((a ^{2+b} ^{2)} ^{2}) (a> 0 ) \int ^{inf ty} _{0} (e ^{-ax} \sin bx)/(x) dx = \arctan (b)/(a) \int ^{inf ty} _{0} (e ^{-ax} - e ^{-bx})/(x) dx =ln (b)/(a) \int ^{inf ty} _{0} (e ^{-ax} - e ^{-bx})/(x) \sin px dx = \arctan (b)/(p) - \arctan (a)/(p) \int ^{inf ty} _{0} (e ^{-ax} - e ^{-bx})/(x) \cos px dx = (1)/(2) ln (b ^{2+p} ^{2})/(a ^{2+p} ^{2}) \int ^{inf ty} _{0} (e ^{-ax} ( 1- \cos x))/(x ^{2}) dx = \arccot a - (a)/(2) ln ( (1)/(a ^{2}) +1 ) \int ^{inf ty} _{-inf ty} e ^{a x ^{4+b} x ^{3+c} x ^{2+d} x + f} dx = e ^{f} \sum ^{inf ty} _{n,m,p =0} (b ^{4n})/(( 4 n)!) (c ^{2m})/(( 2 m)!) (d ^{4p})/(( 4 p)!) (\Gamma( 3 n + m + p + (14 ))/())/(a ^{3 n + m + p + (14)/()}) (appears in several models of extended superstring theory in higher dimensions)\int ^{2 \pi} _{0} e ^{x} d \theta = 2 \pi I _{0(x)} (is the modified Bessel function of the first kind)\int ^{2 \pi} _{0} e ^{x} d \theta = 2 \pi I _{0} \left( \sqrt{x ^{2} + y ^{2}} \right) \int ^{inf ty (x ^{s-1})/(e ^{x/z-1})} _{0} dx = \operatorname{Li} _{s} (z)\Gamma(s),

where

\operatorname{Li} _{s} (z) is the Polylogarithm.

\int ^{inf ty (\sin mx)/(e ^{2} -1)} _{0} dx = (1)/(4) \coth (m)/(2) - (1)/(2 m) \int ^{inf ty} _{0} e ^{-x} 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,$$\backslash int\_0^\; e^\; d\backslash phi\; =\; 2\; \backslash pi\; \backslash delta\_\; \backslash qquad\backslash textm,n\backslash in\backslash mathbb$$where

\delta _{m,n} is the Kronecker delta.

See also

• Gradshteyn and Ryzhik

References

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.https://www.isroset.org/pdf_paper_view.php?paper_id=3100&1-ISROSET-IJSRMSS-08692.pdf

Further reading

• Book: Moll, Victor Hugo . Victor Hugo Moll . Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume I . I . 1 . Series: Monographs and Research Notes in Mathematics . . 2014-11-12 . 978-1-48225-651-2 . 2016-02-12.
• Book: Moll, Victor Hugo . Victor Hugo Moll . Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume II . II . 1 . Series: Monographs and Research Notes in Mathematics . . 2015-10-27 . 978-1-48225-653-6 . 2016-02-12.
• 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
• Web site: Victor Hugo . Moll . Victor Hugo Moll . List with the formulas and proofs in GR . 2016-02-12.