![](https://math-os.com/wp-content/uploads/2018/07/icone-Math-OS-Article-Superieur-205-205.png)
Il existe deux formules de moyenne pour les intégrales.
Elles sont attribuées au mathématicien Pierre-Ossian Bonnet (1819 – 1892).
On se propose de les énoncer, de les établir et d’en donner des exemples significatifs d’utilisation.
Des versions plus fortes de ces résultats existent dans la littérature, mais on a choisi ici de privilégier la simplicité à la généralité.
![](https://math-os.com/wp-content/uploads/2023/08/Bonnet-Pierre-Ossian-787x1024.jpg)
Première formule de la moyenne
On suppose que et
sont des applications continues d’un segment
dans
Proposition 1
Si est positive, alors il existe
tel que :
Notons respectivement et
le minimum et le maximum de
Pour tout
:
![Rendered by QuickLaTeX.com \psi](https://math-os.com/wp-content/ql-cache/quicklatex.com-0c5a80625177ba20ba5092bb7fdca6f3_l3.png)
![Rendered by QuickLaTeX.com \psi](https://math-os.com/wp-content/ql-cache/quicklatex.com-0c5a80625177ba20ba5092bb7fdca6f3_l3.png)
![Rendered by QuickLaTeX.com \int_{a}^{b}\thinspace\psi\left(t\right)\thinspace dt>0](https://math-os.com/wp-content/ql-cache/quicklatex.com-1051c95554286da8286949276483d2de_l3.png)
![Rendered by QuickLaTeX.com \psi](https://math-os.com/wp-content/ql-cache/quicklatex.com-0c5a80625177ba20ba5092bb7fdca6f3_l3.png)
![Rendered by QuickLaTeX.com c\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c312063890bb007d12a5003b1c15733a_l3.png)
![Rendered by QuickLaTeX.com \psi](https://math-os.com/wp-content/ql-cache/quicklatex.com-0c5a80625177ba20ba5092bb7fdca6f3_l3.png)
Remarque 1
On peut remplacer l’hypothèse positive par
de signe constant. Si
est à valeurs négatives, il suffit en effet d’appliquer ce qui précède au couple
En outre, la continuité de n’a pas servi : on peut se contenter de supposer que
est continue par morceaux, ou même seulement Riemann-intégrable.
Remarque 2
En choisissant pour la fonction constante égale à 1, on obtient pour toute application continue
, l’existence d’un réel
tel que :
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
Seconde formule de la moyenne
On suppose toujours que et
sont des applications d’un segment
dans
Proposition 2
Si est décroissante et positive et si
est continue par morceaux (ou seulement Riemann-intégrable), alors il existe
tel que :
Preuve simplifiée n° 1
➡ Supposons en outre
L’application :
Remarque
Sans hypothèse de positivité pour on peut considérer plutôt :
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
![Rendered by QuickLaTeX.com \psi)](https://math-os.com/wp-content/ql-cache/quicklatex.com-8214cde7404ab04785bdb6d196bb0fd4_l3.png)
![Rendered by QuickLaTeX.com K](https://math-os.com/wp-content/ql-cache/quicklatex.com-88ce0c3f3181c87409711287222657dd_l3.png)
![Rendered by QuickLaTeX.com c\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c312063890bb007d12a5003b1c15733a_l3.png)
![Rendered by QuickLaTeX.com \int_{a}^{b}\varphi\left(t\right)\psi\left(t\right)\thinspace dt=K\left(c\right);](https://math-os.com/wp-content/ql-cache/quicklatex.com-a6717b756f015522dafefbdf6e602e4c_l3.png)
Preuve simplifiée n° 2
➡ Supposons en outre de classe
et
continue.
Posons Comme
est continue,
est de classe
et
Après intégration par parties :
![Rendered by QuickLaTeX.com \varphi'](https://math-os.com/wp-content/ql-cache/quicklatex.com-40e80ca7e11fc2751aedee17bbc825ac_l3.png)
![Rendered by QuickLaTeX.com \alpha\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-4f1b9b0617758e21d4a54c8ee93a186b_l3.png)
![Rendered by QuickLaTeX.com \varphi\left(a\right)=0,](https://math-os.com/wp-content/ql-cache/quicklatex.com-cdc1ae4a22ae683ab0859b2a802a8768_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
![Rendered by QuickLaTeX.com \varphi=0](https://math-os.com/wp-content/ql-cache/quicklatex.com-07b3663a95c8047dbebcfbcdb1e9b708_l3.png)
Sinon :
![Rendered by QuickLaTeX.com \dfrac{\varphi\left(b\right)}{\varphi\left(a\right)}\in\left[0,1\right],](https://math-os.com/wp-content/ql-cache/quicklatex.com-4ccb12bb9ae4a302f49736c50911b958_l3.png)
![Rendered by QuickLaTeX.com \Psi\left(b\right)](https://math-os.com/wp-content/ql-cache/quicklatex.com-728364dcd63d26b25d21bbb730e30990_l3.png)
![Rendered by QuickLaTeX.com \Psi\left(\alpha\right).](https://math-os.com/wp-content/ql-cache/quicklatex.com-f04321655cd69a3b144c43143d60a36b_l3.png)
![Rendered by QuickLaTeX.com c\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c312063890bb007d12a5003b1c15733a_l3.png)
Preuve simplifiée n° 3
➡ Supposons en outre continue.
Etant donné notons pour tout
:
![Rendered by QuickLaTeX.com t_{j}](https://math-os.com/wp-content/ql-cache/quicklatex.com-299c72e812a0dfb5162d049efade10ba_l3.png)
![Rendered by QuickLaTeX.com \left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c1c9b62f4ffe493505d6566104a9b311_l3.png)
![Rendered by QuickLaTeX.com t_{n,j}](https://math-os.com/wp-content/ql-cache/quicklatex.com-c227a694e0a6a5a7469b97cde11be5f8_l3.png)
![Rendered by QuickLaTeX.com n](https://math-os.com/wp-content/ql-cache/quicklatex.com-a44d662e2fcd865f31268b1147c8a4be_l3.png)
()
Pour tout :
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
![Rendered by QuickLaTeX.com \epsilon>0](https://math-os.com/wp-content/ql-cache/quicklatex.com-c915c587cdc432e9cd8f4696d6d6ea14_l3.png)
![Rendered by QuickLaTeX.com \delta>0](https://math-os.com/wp-content/ql-cache/quicklatex.com-41b37570ebbb37afeeca331777787074_l3.png)
![Rendered by QuickLaTeX.com n\geqslant\dfrac{b-a}{\delta}](https://math-os.com/wp-content/ql-cache/quicklatex.com-f68e79c354d1570d1f51446c2568b4cb_l3.png)
![Rendered by QuickLaTeX.com \left(\star\right)](https://math-os.com/wp-content/ql-cache/quicklatex.com-d928fdc3003a591d91c4792b57b630c9_l3.png)
Par ailleurs, en notant
![Rendered by QuickLaTeX.com m](https://math-os.com/wp-content/ql-cache/quicklatex.com-c4ea8ba79d799289ca6a2d4c5d0291ea_l3.png)
![Rendered by QuickLaTeX.com M](https://math-os.com/wp-content/ql-cache/quicklatex.com-196f35297f3133f160659c05d5e2fdd5_l3.png)
![Rendered by QuickLaTeX.com \Psi.](https://math-os.com/wp-content/ql-cache/quicklatex.com-397588b2d3b986cdba038d8ed587ae33_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
![Rendered by QuickLaTeX.com \varphi](https://math-os.com/wp-content/ql-cache/quicklatex.com-16671bf72d97106a34b0680eebd2ffe0_l3.png)
![Rendered by QuickLaTeX.com j\in\left\llbracket 0,n-1\right\rrbracket](https://math-os.com/wp-content/ql-cache/quicklatex.com-25008a94ff6b313cab83b39416e2bb8c_l3.png)
![Rendered by QuickLaTeX.com (\star)](https://math-os.com/wp-content/ql-cache/quicklatex.com-cd893208d20baa5812e1ecb543a5a081_l3.png)
![Rendered by QuickLaTeX.com \varphi\left(a\right)=0](https://math-os.com/wp-content/ql-cache/quicklatex.com-7923e56ec9ca1e2accfbc74323f908b0_l3.png)
![Rendered by QuickLaTeX.com \varphi=0](https://math-os.com/wp-content/ql-cache/quicklatex.com-07b3663a95c8047dbebcfbcdb1e9b708_l3.png)
![Rendered by QuickLaTeX.com c\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c312063890bb007d12a5003b1c15733a_l3.png)
Preuve du cas général
Rappel : on suppose décroissante positive et
continue par morceaux (ou seulement Riemann-intégrable).
Nous allons constater que la relation de la preuve simplifiée n° 3 reste vraie. En gros, on perd l’uniforme continuité de
mais on a toujours celle de
![Rendered by QuickLaTeX.com n\geqslant1](https://math-os.com/wp-content/ql-cache/quicklatex.com-7f7b4182cdcb25662bd6fa9413f61256_l3.png)
![Rendered by QuickLaTeX.com \delta\in\left]0,b-a\right[](https://math-os.com/wp-content/ql-cache/quicklatex.com-cb6da155b8cc91000ad56897678373a4_l3.png)
![Rendered by QuickLaTeX.com \Psi^{\star}](https://math-os.com/wp-content/ql-cache/quicklatex.com-8d4b0b8c2cec2cec84421ed8f50822f1_l3.png)
![Rendered by QuickLaTeX.com {\displaystyle \lim_{\delta\rightarrow0}\omega\left(\delta\right)=0.}](https://math-os.com/wp-content/ql-cache/quicklatex.com-3102b9856722bccaaff4ff7fa4a828c9_l3.png)
![Rendered by QuickLaTeX.com \left(\star\right)](https://math-os.com/wp-content/ql-cache/quicklatex.com-d928fdc3003a591d91c4792b57b630c9_l3.png)
Deux exemples d’utilisation
➡ Exemple 1
Soit On considère une application continue
, une application
continue et
périodique et l’on s’intéresse au calcul de :
Pour tout posons :
![Rendered by QuickLaTeX.com x=nt](https://math-os.com/wp-content/ql-cache/quicklatex.com-42244f2fc6cfe34e6c25cfd41236bf4c_l3.png)
![Rendered by QuickLaTeX.com x=s+kT](https://math-os.com/wp-content/ql-cache/quicklatex.com-f604bda135d79f5d444ddaeeacc5b5b0_l3.png)
![Rendered by QuickLaTeX.com k-](https://math-os.com/wp-content/ql-cache/quicklatex.com-15adf0ac22fab9adc010197a54cee823_l3.png)
![Rendered by QuickLaTeX.com g](https://math-os.com/wp-content/ql-cache/quicklatex.com-d3aa3046c4e90d3971b3677de11903f1_l3.png)
![Rendered by QuickLaTeX.com T-](https://math-os.com/wp-content/ql-cache/quicklatex.com-a8df60e148b1cbe13aaa49c9627a4724_l3.png)
![Rendered by QuickLaTeX.com g](https://math-os.com/wp-content/ql-cache/quicklatex.com-d3aa3046c4e90d3971b3677de11903f1_l3.png)
![Rendered by QuickLaTeX.com k\in\left\llbracket 0,n-1\right\rrbracket ,](https://math-os.com/wp-content/ql-cache/quicklatex.com-a5a74700e14fa060e69d6a2a70bed3da_l3.png)
![Rendered by QuickLaTeX.com c_{k}\in\left[0,T\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-d63bdefba79f85bfcdebad4f69b09711_l3.png)
![Rendered by QuickLaTeX.com R_{n}](https://math-os.com/wp-content/ql-cache/quicklatex.com-46c2e1743bd5fcf251fff83d2b3e6635_l3.png)
![Rendered by QuickLaTeX.com f](https://math-os.com/wp-content/ql-cache/quicklatex.com-4de78b071f57702a0dfd4345a28e8840_l3.png)
![Rendered by QuickLaTeX.com \left[0,T\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-662710167665f3fe8fec19179f2eb7fc_l3.png)
![Rendered by QuickLaTeX.com g](https://math-os.com/wp-content/ql-cache/quicklatex.com-d3aa3046c4e90d3971b3677de11903f1_l3.png)
![Rendered by QuickLaTeX.com \lambda\in\mathbb{R}](https://math-os.com/wp-content/ql-cache/quicklatex.com-6439e9f8fb0de2ef5854c1e2891b2f32_l3.png)
![Rendered by QuickLaTeX.com \gamma:t\mapsto\lambda+g\left(t\right)](https://math-os.com/wp-content/ql-cache/quicklatex.com-214831764979ac7c1d53ed30b317abb0_l3.png)
![Rendered by QuickLaTeX.com \lambda\geqslant-\inf_{\left[0,T\right]}g).](https://math-os.com/wp-content/ql-cache/quicklatex.com-24e64529f6becd0e3a9ab6ca405ad653_l3.png)
Remarque
Le lecteur pourra démontrer en exercice une formule un peu plus générale, à savoir que si est définie sur
(au lieu de
on obtient en fin de compte :
A titre d’exemple :
![Rendered by QuickLaTeX.com t\mapsto\sin(nt)](https://math-os.com/wp-content/ql-cache/quicklatex.com-0d12fbf033b07fbe5e2cf1574311d20f_l3.png)
![Rendered by QuickLaTeX.com \pi-](https://math-os.com/wp-content/ql-cache/quicklatex.com-a65deca6d10e227227e583ca05490bf8_l3.png)
![Rendered by QuickLaTeX.com \dfrac{2}{\pi}](https://math-os.com/wp-content/ql-cache/quicklatex.com-3ebaa96215c1eebf60163152ef43dff1_l3.png)
➡ Exemple 2
Soit continue telle que l’intégrale
soit semi-convergente. Pour tout
posons :
![Rendered by QuickLaTeX.com F](https://math-os.com/wp-content/ql-cache/quicklatex.com-6ac0241c2e7319cb42a4efe8e4bc0710_l3.png)
est bien défini par hypothèse. Et pour
l’intégrale
existe d’après le critère de Cauchy. En effet, étant donné
il existe
tel que pour tout
:
![Rendered by QuickLaTeX.com b>a\geqslant A,](https://math-os.com/wp-content/ql-cache/quicklatex.com-74bb9c48bd8d43adba73f687ad6ece24_l3.png)
![Rendered by QuickLaTeX.com c\in\left[a,b\right]](https://math-os.com/wp-content/ql-cache/quicklatex.com-c312063890bb007d12a5003b1c15733a_l3.png)
![Rendered by QuickLaTeX.com n\in\mathbb{N}](https://math-os.com/wp-content/ql-cache/quicklatex.com-40c78fca23fcd2c6858fe61d5e522aaa_l3.png)
![Rendered by QuickLaTeX.com x\in\left[0,+\infty\right[](https://math-os.com/wp-content/ql-cache/quicklatex.com-b92b099d80936a0d2b52e421ce034e18_l3.png)
![Rendered by QuickLaTeX.com f](https://math-os.com/wp-content/ql-cache/quicklatex.com-4de78b071f57702a0dfd4345a28e8840_l3.png)
![Rendered by QuickLaTeX.com F_{n}](https://math-os.com/wp-content/ql-cache/quicklatex.com-4e1caf606fed96fe4d61d059fa07badb_l3.png)
![Rendered by QuickLaTeX.com \left(F_{n}\right)_{n\geqslant0}](https://math-os.com/wp-content/ql-cache/quicklatex.com-a9ee60e5e6bba446cfee9fb4167cea43_l3.png)
![Rendered by QuickLaTeX.com \left[0,+\infty\right[](https://math-os.com/wp-content/ql-cache/quicklatex.com-9ad4ca5cdabd05314823703f5a2d6f5b_l3.png)
![Rendered by QuickLaTeX.com F,](https://math-os.com/wp-content/ql-cache/quicklatex.com-86a7ee2db61c3ba27151b857242cd718_l3.png)
![Rendered by QuickLaTeX.com F](https://math-os.com/wp-content/ql-cache/quicklatex.com-6ac0241c2e7319cb42a4efe8e4bc0710_l3.png)
![Rendered by QuickLaTeX.com \left(p,q\right)\in\mathbb{N}^{2}](https://math-os.com/wp-content/ql-cache/quicklatex.com-59390bb82505c498c8ef7b3cbafc36c2_l3.png)
![Rendered by QuickLaTeX.com q>p\geqslant A](https://math-os.com/wp-content/ql-cache/quicklatex.com-3b3fa6c9422c03356a84c035e68b71c0_l3.png)
![Rendered by QuickLaTeX.com x\in\left[0,+\infty\right[](https://math-os.com/wp-content/ql-cache/quicklatex.com-b92b099d80936a0d2b52e421ce034e18_l3.png)
![Rendered by QuickLaTeX.com c\in\left[p,q\right].](https://math-os.com/wp-content/ql-cache/quicklatex.com-e73ccf1dcf4ad7071b5ce622a98189b1_l3.png)
Un cas particulier célèbre est celui où :
![Rendered by QuickLaTeX.com \int_{0}^{+\infty}\dfrac{\sin\left(t\right)}{t}\thinspace dt](https://math-os.com/wp-content/ql-cache/quicklatex.com-82a7a114a535bfde6583f63e9cdcb1d3_l3.png)
![Rendered by QuickLaTeX.com x>0](https://math-os.com/wp-content/ql-cache/quicklatex.com-72a0e45805dc43ad27f96050e3d74c14_l3.png)
![Rendered by QuickLaTeX.com C\in\mathbb{R}](https://math-os.com/wp-content/ql-cache/quicklatex.com-9294353c164cba0080b3811d19c8fd6b_l3.png)
![Rendered by QuickLaTeX.com C=\dfrac{\pi}{2}.](https://math-os.com/wp-content/ql-cache/quicklatex.com-1737acde17cbbc997b63db631092a97d_l3.png)
![Rendered by QuickLaTeX.com 0](https://math-os.com/wp-content/ql-cache/quicklatex.com-d43ee783c9dd19196231e380f0830a9f_l3.png)
![Rendered by QuickLaTeX.com x\mapsto\int_{0}^{+\infty}e^{-xt}\thinspace\dfrac{\sin\left(t\right)}{t}\thinspace dt,](https://math-os.com/wp-content/ql-cache/quicklatex.com-a7f2e9428dfd9aee34243d1fdc739556_l3.png)
Remarque 1
Dans ce calcul, il était essentiel de prouver la continuité de en
On aurait pu établir la continuité de
sur
à moindre frais, car il suffit de prouver la convergence uniforme de la suite
sur
pour tout
ce qui s’obtient par une majoration directe. En effet, pour tout
et tout
:
![Rendered by QuickLaTeX.com \int_0^{+\infty}\frac{\sin(t)}{t}\,dt](https://math-os.com/wp-content/ql-cache/quicklatex.com-bad6949874183b1bd3b0b920e43600e9_l3.png)
Remarque 2
Si, dans l’énoncé de l’exemple 2, l’hypothèse de semi-convergence de l’intégrale impropre est remplacée par l’hypothèse plus forte d’absolue convergence, la continuité de
est beaucoup plus simple à établir. En effet, pour tout
et pour tout
:
Pour toutes questions ou remarques, merci d’utiliser le formulaire de contact.
L’exemple 1 est une question classique mais qui est super traitée .
Super et continuez !
C’est un plaisir de lire vos articles