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Lista de integrais de funções exponenciais
A lista seguinte contém
integrais
de
funções exponenciais
.
Integral Exponencial:
∫
e
c
x
d
x
=
1
c
e
c
x
{\displaystyle \int e^{cx}\;dx={\frac {1}{c}e^{cx}
∫
a
c
x
d
x
=
1
c
ln
a
a
c
x
(para
a
>
0
,
a
≠
1
)
{\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}a^{cx}\qquad {\mbox{(para }a>0,{\mbox{ }a\neq 1{\mbox{)}
∫
x
e
c
x
d
x
=
e
c
x
c
2
(
c
x
−
1
)
{\displaystyle \int xe^{cx}\;dx={\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}\;dx=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
{\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}x^{n}e^{cx}-{\frac {n}{c}\int x^{n-1}e^{cx}dx}
∫
e
c
x
d
x
x
=
ln
|
x
|
+
∑
i
=
1
∞
(
c
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {e^{cx}\;dx}{x}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}{i\cdot i!}
∫
e
c
x
d
x
x
n
=
1
n
−
1
(
−
e
c
x
x
n
−
1
+
c
∫
e
c
x
d
x
x
n
−
1
)
(para
n
≠
1
)
{\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}={\frac {1}{n-1}\left(-{\frac {e^{cx}{x^{n-1}+c\int {\frac {e^{cx}dx}{x^{n-1}\right)\qquad {\mbox{(para }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\;dx={\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\;dx={\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\;dx={\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\;dx={\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\;dx}
∫
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\;dx={\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\;dx}
∫
x
e
c
x
2
d
x
=
1
2
c
e
c
x
2
{\displaystyle \int xe^{cx^{2}\;dx={\frac {1}{2c}\;e^{cx^{2}
∫
1
σ
2
π
e
−
(
x
−
μ
)
2
/
2
σ
2
d
x
=
1
2
σ
(
1
+
erf
x
−
μ
σ
2
)
{\displaystyle \int {1 \over \sigma {\sqrt {2\pi }\,e^{-{(x-\mu )^{2}/2\sigma ^{2}\;dx={\frac {1}{2\sigma }(1+{\mbox{erf}\,{\frac {x-\mu }{\sigma {\sqrt {2})}
∫
e
x
2
d
x
=
e
x
2
(
∑
r
=
1
n
1
2
n
x
2
n
−
1
)
+
2
n
−
1
2
n
∫
e
x
2
d
x
x
2
n
{\displaystyle \int e^{x^{2}\,dx=e^{x^{2}\left(\sum _{r=1}^{n}{\frac {1}{2^{n}x^{2n-1}\right)+{\frac {2n-1}{2^{n}\int {\frac {e^{x^{2}\;dx}{x^{2n}
∫
−
∞
∞
e
−
a
x
2
d
x
=
π
a
{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}\,dx={\sqrt {\pi \over a}
