Fie o mulțime nevidă
A
{\displaystyle A}
[
a
,
b
]
⊂
R
{\displaystyle [a,b]\subset \mathbb {R} }
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
y
∈
A
{\displaystyle y\in A}
x
↦
f
(
x
,
y
)
∈
R
{\displaystyle x\mapsto f(x,y)\in \mathbb {R} }
integrabilă Riemann .
Funcția definită prin:
F
:
A
→
R
,
F
(
y
)
=
∫
a
b
f
(
x
,
y
)
d
x
{\displaystyle F:A\to \mathbb {R} ,\;F(y)=\int _{a}^{b}f(x,y)dx}
se numește integrală cu parametru.
Dacă
f
:
[
a
,
b
]
×
A
↦
R
{\displaystyle f:[a,b]\times A\mapsto \mathbb {R} }
F
(
y
)
=
∫
a
b
f
(
x
,
y
)
d
x
{\displaystyle F(y)=\int _{a}^{b}f(x,y)dx}
funcție continuă .
Fie
f
:
[
a
,
b
]
×
(
c
,
d
)
↦
R
{\displaystyle f:[a,b]\times (c,d)\mapsto \mathbb {R} }
∂
f
∂
y
{\displaystyle {\frac {\partial f}{\partial y}}}
[
a
,
b
]
×
(
c
,
d
)
.
{\displaystyle [a,b]\times (c,d).}
F
(
y
)
=
∫
a
b
f
(
x
,
y
)
d
x
{\displaystyle F(y)=\int _{a}^{b}f(x,y)dx}
F
′
(
y
)
=
∫
a
b
∂
f
∂
y
(
x
,
y
)
d
x
,
∀
y
∈
(
c
,
d
)
.
{\displaystyle F'(y)=\int _{a}^{b}{\frac {\partial f}{\partial y}}(x,y)dx,\;\forall y\in (c,d).}
![{\displaystyle [a,b]\subset \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a659536067aaaac2db1c44613a09a715f0cf7246)
![{\displaystyle f:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b592d102ccd1ba134d401c5b3ea177baaba3ffac)
![{\displaystyle f:[a,b]\times A\mapsto \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/beb75de42e10bea841f2c619e9f800f79ca81ed0)
![{\displaystyle f:[a,b]\times (c,d)\mapsto \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47fa8a03d3f98e091257c01a7359147da5466d6)
![{\displaystyle [a,b]\times (c,d).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/759250c09ff47de4b2a9f2a472631b68ab2ed839)
