===== Function integral on ℝⁿ =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $p\in \mathbb N$ |
| @#FFBB00: definiendum | @#FFBB00: $I^p: (\mathbb R^p\to\overline{\mathbb R})\to\overline{\mathbb R}$ |
| @#FFBB00: definiendum | @#FFBB00: $I^p(f):=\int_{\mathbb R^p}\ f\ \mathrm d\lambda^p$ |

==== Discussion ====
Because the integral above coincides with the Lebesgue–Stieltjes integral for the monotone function $F(x):=x$, we'll also denote $I^p(f)$ by $\int_{\mathbb R^p}\ f(x)\ \mathrm dx^p$ with the argument $x\in \mathbb R^p$ of $f$ becoming a dummy index.

=== Theorems ===
For $f:X\to \mathbb R$...differentiable  and $f'$...[[Bounded function|bounded]], we have

^ $ \int_a^b\dfrac{{\mathrm d}f}{{\mathrm d}x}{\mathrm d}x = f(b)-f(a) $ ^
^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^

For $f$ convex and 

$\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$

^ $\dfrac{f(a) + f(b)}{2} \ge \langle f\rangle_{[a,b]} \ge f \left(\dfrac{a+b}{2}\right) $ ^

See references.

== Kernel of he integral ==
A linear combination of functions that are zero under an integral are again zero.

Special case

$$\int_{-a}^a E(x) \left( \dfrac{1}{2} + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) \,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$

e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$:

$$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$

$$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$

=== References ===
Wikipedia:
[[https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality|Hermite–Hadamard inequality]]

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=== Subset of ===
[[Function integral]]
=== Context ===
[[Lebesgue measure]]
=== Related ===
[[Integral over a subset]], [[Fréchet derivative]]