Seznam integralov eksponentnih funkcij
Seznam integralov eksponentnih funkcij vsebuje integrale eksponentnih funkcij .
Nedoločeni integrali so primitivne funkcije . Aditivno konstanto lahko dodamo na desni strani vsakega izmed obrazcev, tukaj so te konstante izpuščene zaradi enostavnosti.
∫
e
x
d
x
=
e
x
{\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}
∫
e
c
x
d
x
=
1
c
e
c
x
{\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}e^{cx}
∫
a
c
x
d
x
=
1
c
⋅
ln
a
a
c
x
{\displaystyle \int a^{cx}\;\mathrm {d} x={\frac {1}{c\cdot \ln a}a^{cx}
a
>
0
,
a
≠
1
{\displaystyle a>0,\ a\neq 1}
∫
x
e
c
x
d
x
=
e
c
x
c
2
(
c
x
−
1
)
{\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}{c^{2}(cx-1)}
∫
x
2
e
c
x
d
x
=
e
c
x
(
x
2
c
−
2
x
c
2
+
2
c
3
)
{\displaystyle \int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}{c}-{\frac {2x}{c^{2}+{\frac {2}{c^{3}\right)}
∫
x
n
e
c
x
d
x
=
1
c
x
n
e
c
x
−
n
c
∫
x
n
−
1
e
c
x
d
x
=
(
∂
∂
c
)
n
e
c
x
c
{\displaystyle \int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}x^{n}e^{cx}-{\frac {n}{c}\int x^{n-1}e^{cx}\mathrm {d} x=\left({\frac {\partial }{\partial c}\right)^{n}{\frac {e^{cx}{c}
∫
e
c
x
x
d
x
=
ln
|
x
|
+
∑
n
=
1
∞
(
c
x
)
n
n
⋅
n
!
{\displaystyle \int {\frac {e^{cx}{x}\;\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}{n\cdot n!}
∫
e
c
x
x
n
d
x
=
1
n
−
1
(
−
e
c
x
x
n
−
1
+
c
∫
e
c
x
x
n
−
1
d
x
)
(za
n
≠
1
)
{\displaystyle \int {\frac {e^{cx}{x^{n}\;\mathrm {d} x={\frac {1}{n-1}\left(-{\frac {e^{cx}{x^{n-1}+c\int {\frac {e^{cx}{x^{n-1}\,\mathrm {d} x\right)\qquad {\mbox{(za }n\neq 1{\mbox{)}
∫
e
c
x
ln
x
d
x
=
1
c
e
c
x
ln
|
x
|
−
Ei
(
c
x
)
{\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}
∫
e
c
x
sin
b
x
d
x
=
e
c
x
c
2
+
b
2
(
c
sin
b
x
−
b
cos
b
x
)
{\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}{c^{2}+b^{2}(c\sin bx-b\cos bx)}
∫
e
c
x
cos
b
x
d
x
=
e
c
x
c
2
+
b
2
(
c
cos
b
x
+
b
sin
b
x
)
{\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}{c^{2}+b^{2}(c\cos bx+b\sin bx)}
∫
e
c
x
sin
n
x
d
x
=
e
c
x
sin
n
−
1
x
c
2
+
n
2
(
c
sin
x
−
n
cos
x
)
+
n
(
n
−
1
)
c
2
+
n
2
∫
e
c
x
sin
n
−
2
x
d
x
{\displaystyle \int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x}
∫
e
c
x
cos
n
x
d
x
=
e
c
x
cos
n
−
1
x
c
2
+
n
2
(
c
cos
x
+
n
sin
x
)
+
n
(
n
−
1
)
c
2
+
n
2
∫
e
c
x
cos
n
−
2
x
d
x
{\displaystyle \int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x}
∫
x
e
c
x
2
d
x
=
1
2
c
e
c
x
2
{\displaystyle \int xe^{cx^{2}\;\mathrm {d} x={\frac {1}{2c}\;e^{cx^{2}
∫
e
−
c
x
2
d
x
=
π
4
c
erf
(
c
x
)
{\displaystyle \int e^{-cx^{2}\;\mathrm {d} x={\sqrt {\frac {\pi }{4c}\operatorname {erf} ({\sqrt {c}x)}
∫
x
e
−
c
x
2
d
x
=
−
1
2
c
e
−
c
x
2
{\displaystyle \int xe^{-cx^{2}\;\mathrm {d} x=-{\frac {1}{2c}e^{-cx^{2}
∫
1
σ
2
π
e
−
(
x
−
μ
)
2
/
2
σ
2
d
x
=
−
1
2
(
erf
−
x
+
μ
σ
2
)
{\displaystyle \int {1 \over \sigma {\sqrt {2\pi }\,e^{-{(x-\mu )^{2}/2\sigma ^{2}\;\mathrm {d} x=-{\frac {1}{2}\left(\operatorname {erf} \,{\frac {-x+\mu }{\sigma {\sqrt {2}\right)\ \ \ }
funkcija napake )
∫
e
x
2
d
x
=
e
x
2
(
∑
j
=
0
n
−
1
c
2
j
1
x
2
j
+
1
)
+
(
2
n
−
1
)
c
2
n
−
2
∫
e
x
2
x
2
n
d
x
velja za
n
>
0
,
{\displaystyle \int e^{x^{2}\,\mathrm {d} x=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}\;\mathrm {d} x\quad {\mbox{velja za }n>0,}
where
c
2
j
=
1
⋅
3
⋅
5
⋯
(
2
j
−
1
)
2
j
+
1
=
(
2
j
)
!
j
!
2
2
j
+
1
.
{\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}={\frac {(2j)\,!}{j!\,2^{2j+1}\ .}
∫
x
x
⋅
⋅
x
⏟
m
d
x
=
∑
n
=
0
m
(
−
1
)
n
(
n
+
1
)
n
−
1
n
!
Γ
(
n
+
1
,
−
ln
x
)
+
∑
n
=
m
+
1
∞
(
−
1
)
n
a
m
n
Γ
(
n
+
1
,
−
ln
x
)
(za
x
>
0
)
{\displaystyle {\int \underbrace {x^{x^{\cdot ^{\cdot ^{x} _{m}\,dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}{n!}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(za }x>0{\mbox{)}
where
a
m
n
=
{
1
kadar je
n
=
0
,
1
n
!
kadar je
m
=
1
,
1
n
∑
j
=
1
n
j
a
m
,
n
−
j
a
m
−
1
,
j
−
1
v ostalih primerih
{\displaystyle a_{mn}={\begin{cases}1&{\text{kadar je }n=0,\\{\frac {1}{n!}&{\text{kadar je }m=1,\\{\frac {1}{n}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{v ostalih primerih }\end{cases}
in
Γ
(
x
,
y
)
{\displaystyle \Gamma (x,y)}
∫
1
a
e
λ
x
+
b
d
x
=
x
b
−
1
b
λ
ln
(
a
e
λ
x
+
b
)
{\displaystyle \int {\frac {1}{ae^{\lambda x}+b}\;\mathrm {d} x={\frac {x}{b}-{\frac {1}{b\lambda }\ln \left(ae^{\lambda x}+b\right)\,}
b
≠
0
{\displaystyle b\neq 0}
λ
≠
0
{\displaystyle \lambda \neq 0}
a
e
λ
x
+
b
>
0
.
{\displaystyle ae^{\lambda x}+b>0\,.}
∫
e
2
λ
x
a
e
λ
x
+
b
d
x
=
1
a
2
λ
[
a
e
λ
x
+
b
−
b
ln
(
a
e
λ
x
+
b
)
]
{\displaystyle \int {\frac {e^{2\lambda x}{ae^{\lambda x}+b}\;\mathrm {d} x={\frac {1}{a^{2}\lambda }\left[ae^{\lambda x}+b-b\ln \left(ae^{\lambda x}+b\right)\right]\,}
a
≠
0
{\displaystyle a\neq 0}
λ
≠
0
{\displaystyle \lambda \neq 0}
a
e
λ
x
+
b
>
0
{\displaystyle ae^{\lambda x}+b>0\,}
∫
0
1
e
x
⋅
ln
a
+
(
1
−
x
)
⋅
ln
b
d
x
=
∫
0
1
(
a
b
)
x
⋅
b
d
x
=
∫
0
1
a
x
⋅
b
1
−
x
d
x
=
a
−
b
ln
a
−
ln
b
{\displaystyle \int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\;\mathrm {d} x=\int _{0}^{1}\left({\frac {a}{b}\right)^{x}\cdot b\;\mathrm {d} x=\int _{0}^{1}a^{x}\cdot b^{1-x}\;\mathrm {d} x={\frac {a-b}{\ln a-\ln b}
a
>
0
,
b
>
0
,
a
≠
b
{\displaystyle a>0,\ b>0,\ a\neq b}
∫
0
∞
e
a
x
d
x
=
1
|
a
|
(
a
<
0
)
{\displaystyle \int _{0}^{\infty }e^{ax}\,\mathrm {d} x={\frac {1}{|a|}\quad (a<0)}
∫
0
∞
e
−
a
x
2
d
x
=
1
2
π
a
(
a
>
0
)
{\displaystyle \int _{0}^{\infty }e^{-ax^{2}\,\mathrm {d} x={\frac {1}{2}{\sqrt {\pi \over a}\quad (a>0)}
∫
−
∞
∞
e
−
a
x
2
d
x
=
π
a
(
a
>
0
)
{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}\,\mathrm {d} x={\sqrt {\pi \over a}\quad (a>0)}
∫
−
∞
∞
e
−
a
x
2
e
−
2
b
x
d
x
=
π
a
e
b
2
a
(
a
>
0
)
{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}e^{-2bx}\,\mathrm {d} x={\sqrt {\frac {\pi }{a}e^{\frac {b^{2}{a}\quad (a>0)}
∫
−
∞
∞
x
e
−
a
(
x
−
b
)
2
d
x
=
b
π
a
{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}
∫
−
∞
∞
x
2
e
−
a
x
2
d
x
=
1
2
π
a
3
(
a
>
0
)
{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}\,\mathrm {d} x={\frac {1}{2}{\sqrt {\pi \over a^{3}\quad (a>0)}
∫
0
∞
x
n
e
−
a
x
2
d
x
=
{
1
2
Γ
(
n
+
1
2
)
/
a
n
+
1
2
(
n
>
−
1
,
a
>
0
)
(
2
k
−
1
)
!
!
2
k
+
1
a
k
π
a
(
n
=
2
k
,
k
integer
,
a
>
0
)
k
!
2
a
k
+
1
(
n
=
2
k
+
1
,
k
integer
,
a
>
0
)
{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}\,\mathrm {d} x={\begin{cases}{\frac {1}{2}\Gamma \left({\frac {n+1}{2}\right)/a^{\frac {n+1}{2}&(n>-1,a>0)\\{\frac {(2k-1)!!}{2^{k+1}a^{k}{\sqrt {\frac {\pi }{a}&(n=2k,k\;{\text{integer},a>0)\\{\frac {k!}{2a^{k+1}&(n=2k+1,k\;{\text{integer},a>0)\end{cases}
dvojno fakulteto )
∫
0
∞
x
n
e
−
a
x
d
x
=
{
Γ
(
n
+
1
)
a
n
+
1
(
n
>
−
1
,
a
>
0
)
n
!
a
n
+
1
(
n
=
0
,
1
,
2
,
…
,
a
>
0
)
{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\,\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}&(n=0,1,2,\ldots ,a>0)\\\end{cases}
∫
0
∞
e
−
a
x
sin
b
x
d
x
=
b
a
2
+
b
2
(
a
>
0
)
{\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,\mathrm {d} x={\frac {b}{a^{2}+b^{2}\quad (a>0)}
∫
0
∞
e
−
a
x
cos
b
x
d
x
=
a
a
2
+
b
2
(
a
>
0
)
{\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,\mathrm {d} x={\frac {a}{a^{2}+b^{2}\quad (a>0)}
∫
0
∞
x
e
−
a
x
sin
b
x
d
x
=
2
a
b
(
a
2
+
b
2
)
2
(
a
>
0
)
{\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx\,\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}\quad (a>0)}
∫
0
∞
x
e
−
a
x
cos
b
x
d
x
=
a
2
−
b
2
(
a
2
+
b
2
)
2
(
a
>
0
)
{\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx\,\mathrm {d} x={\frac {a^{2}-b^{2}{(a^{2}+b^{2})^{2}\quad (a>0)}
∫
0
2
π
e
x
cos
θ
d
θ
=
2
π
I
0
(
x
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}
I
0
{\displaystyle I_{0}
Besselova funkcija prve vrste)
∫
0
2
π
e
x
cos
θ
+
y
sin
θ
d
θ
=
2
π
I
0
(
x
2
+
y
2
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}\right)}
Glej tudi
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![{\displaystyle \int {\frac {e^{2\lambda x}{ae^{\lambda x}+b}\;\mathrm {d} x={\frac {1}{a^{2}\lambda }\left[ae^{\lambda x}+b-b\ln \left(ae^{\lambda x}+b\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/278961ab24efa75c218ed9fc570b0f9a7dac350f)
