Chapitres Maths en ECG1
Chapitres Maths en ECG1
Corrigés : Intégration en ECG1
Résumé de cours Exercices Corrigés
Cours en ligne de Maths en ECG1
Corrigés – Intégration
Exercice 1 :
1) L’expression (de la forme
) se primitive en
ainsi
2) Commençons par linéariser On utilise la formule de Moivre-Euler
.
D’où
3) On écrit
![Rendered by QuickLaTeX.com \int_0^{\dfrac12} \dfrac{x}{ \sqrt{1 - x^2}} dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1012a5d7b246b17c3a1d821f6df1351d_l3.png)
![Rendered by QuickLaTeX.com = - \dfrac12 \int_0^{\dfrac12} \dfrac{- 2 x}{\sqrt{ 1 - x^2} } dx.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7adfbff62a2f653e2ddf1713fd57b05c_l3.png)
![Rendered by QuickLaTeX.com x \to \dfrac{- 2 x}{\sqrt{ 1 - x^2} }](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5ced97d6f8dc8ff34391e21bdd2a6f6b_l3.png)
![Rendered by QuickLaTeX.com u' u^{- \dfrac12}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a668cd452525bac5dbeb5bfc3fec77c6_l3.png)
![Rendered by QuickLaTeX.com x \mapsto 2 \sqrt{1 - x^2},](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b01367c92f9f44caa4a094b428c01f19_l3.png)
![Rendered by QuickLaTeX.com - \dfrac12 \int_0^{\dfrac12} \dfrac{- 2 x}{\sqrt{ 1 - x^2} } dx](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1c1180a5320793fcd0a8a918392196dd_l3.png)
![Rendered by QuickLaTeX.com = -\dfrac12 \left[ 2 \sqrt{1 - x^2} \right]_0^{\dfrac12}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-371108fae56e4b6af9df08533c797240_l3.png)
![Rendered by QuickLaTeX.com = - \sqrt{1 - \left( \dfrac12 \right)^2 } + 1.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4d84d3f9999b501349c3ef79a810558b_l3.png)
![Rendered by QuickLaTeX.com u : t \mapsto \dfrac{t^3}{3}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-082dc9049a79d93ebb1611d104dd9abe_l3.png)
![Rendered by QuickLaTeX.com v : t \mapsto \ln \left( t \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-33c96cd34067b45a6db0775952dcffa7_l3.png)
![Rendered by QuickLaTeX.com u](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png)
![Rendered by QuickLaTeX.com v](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png)
![Rendered by QuickLaTeX.com C^1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b1cd5af4f12e712d1a939f8991cb3e04_l3.png)
![Rendered by QuickLaTeX.com \left[ 1 , e \right]](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-62981420726830f0fa33246835b549f8_l3.png)
![Rendered by QuickLaTeX.com \int_1^e t^2 \ln \left( t \right) dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9cf262bc2f898c003d4acce01fd9ca6e_l3.png)
![Rendered by QuickLaTeX.com = \left[ \dfrac{t^3}{3} \ln \left( t \right) \right]_1^e - \int_1^e \dfrac{t^2}{3} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-213fc30ab347dc53fcd1cdb8614a34f2_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{e^3}{3} - \left[ \dfrac{t^3}{9} \right]_1^e](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2c3b5e13ebe1d0c6a8ee4c0e29b7bfc3_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{e^3}{3} - \dfrac{e^3}{9} + \dfrac19](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-c09ece386370d47bc71775bd1fccc105_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{2 e^3}{9} + \dfrac19](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4606b26be33041ad032562447f9e3283_l3.png)
![Rendered by QuickLaTeX.com u : t \mapsto \mathrm{Arctan} \left( t \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4e68d29442400928b4d19ac80eb88afd_l3.png)
![Rendered by QuickLaTeX.com v : t \mapsto \dfrac{t^2}{2}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1d3e7f8bef72cdf257ffb7de9bc0709e_l3.png)
![Rendered by QuickLaTeX.com u](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png)
![Rendered by QuickLaTeX.com v](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png)
![Rendered by QuickLaTeX.com C^1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b1cd5af4f12e712d1a939f8991cb3e04_l3.png)
![Rendered by QuickLaTeX.com \left[ 0 , 1 \right]](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-77aa6afb09320f5b2c43e80903990f50_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12 \mathrm{Arctan} \left( 1 \right) - \int_0^1 \dfrac{t^2 + 1 - 1}{1 + t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ea763672dee7b921234e9580b2c8dbf1_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12 \dfrac{\pi}{4} - \int_0^1 1 dt + \int_0^1 \dfrac{1}{1 + t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-0439e01821a116bb2717a2480784a105_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{\pi}{8} - 1 + \mathrm{Arctan} \left( 1 \right)](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-49cb40b6d4a55ef8482da04ce6bb16af_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{\pi}{8} + \dfrac{\pi}{4} - 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1bb8f27162ff16ee204d9bfa180e97a2_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{3 \pi}{ 8}- 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fb8d5a090233be2df975edc97c05598b_l3.png)
Exercice 2 :
1) Si l’on pose on commence par remplacer
par
on a donc :
On exprime en fonction de
Comme
on a
soit
ainsi
Il nous reste à trouver les bonne bornes : lorsque et lorsque
d’où finalement :
Cette dernière est plus facile à calculer car se primitive en
d’où :
![Rendered by QuickLaTeX.com \dfrac12 \int_1^{100} \dfrac{1}{ \left( y + 1 \right)^2} dy](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-4a30ab63ccc543b662e1d33a3df7de05_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12 \left[ - \dfrac{1}{y + 1} \right]_1^{100}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-af94e453a4074f4739c09af470a04742_l3.png)
![Rendered by QuickLaTeX.com = \dfrac12 \left( - \dfrac{1}{101} + \dfrac12 \right).](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7ca781140bce794a7346e27ed83f89e8_l3.png)
Pour calculer cette intégrale, il faut linéariser On utilise les formules de Moivre-Euler :
.
Ainsi
Exercice 3 :
1) On a
![Rendered by QuickLaTeX.com = \left[ - \dfrac{ \left( b - t \right)^{q + 1} }{q + 1} \right]_a^b](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2699c46df37f178b4dc83734419d3998_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{ \left( b- a \right)^{q + 1} }{q + 1}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f7a05e60025bf10fa27e63aea5f6aa6d_l3.png)
2) On fait une intégration par parties en posant et
Les fonctions
et
sont
sur
et :
![Rendered by QuickLaTeX.com I_{p , q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9117b07ca7ae686d28d6bc36b5f5e0a7_l3.png)
![Rendered by QuickLaTeX.com = \left[ - \dfrac{\left( t - a \right)^p \left( b - t \right)^{q + 1} }{q + 1} \right]_a^b](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-653c25261e026853cd713d760913dd1e_l3.png)
![Rendered by QuickLaTeX.com + \dfrac{p}{q + 1} \int_a^b \left( t -a \right)^{p - 1} \left( b - t \right)^{q + 1} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-7d8a819fd5b74d752a64aff5397a50f7_l3.png)
![Rendered by QuickLaTeX.com \left[ - \dfrac{\left( t - a \right)^p \left( b - t \right)^{q + 1} }{q + 1} \right]_a^b = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2c957322020f76dcb819fe41963bc901_l3.png)
![Rendered by QuickLaTeX.com I_{p , q} = \dfrac{p}{q + 1} I_{p - 1 , q +1}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ac0cb3ba740642f97a7bd6db86335b37_l3.png)
3) Si l’on applique fois (avec
) la relation précédente, on a :
![Rendered by QuickLaTeX.com I_{p , q}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-9117b07ca7ae686d28d6bc36b5f5e0a7_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{p \left( p - 1 \right) \times \cdots \times \left( p - \left( k - 1 \right) \right)}{\left( q + 1 \right)\times \left( q + 2 \right) \times \cdots \times \left( q + k \right)} I_{p - k , q + k}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-b0eac1fee3ae49f5d3da4ceca8cb4e15_l3.png)
En prenant
![Rendered by QuickLaTeX.com k = p,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-f470e8a57a4ce9ff001d70911cba3ed0_l3.png)
Comme d’où finalement
Exercice 4 :
1) On calcule
![Rendered by QuickLaTeX.com I_{n + 1 } - I_n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-78915a198b6c6e8ca0a61d2c4ea926d1_l3.png)
![Rendered by QuickLaTeX.com = \int_0^1 \dfrac{t^{n + 1}}{1 + t^2} dt - \int_0^1 \dfrac{t^n}{1 + t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-53aa0aa59dc9dff7f570231fc8b899b9_l3.png)
![Rendered by QuickLaTeX.com = \int_0^1 t^n \dfrac{t - 1}{ 1 + t^2} dt.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-fb52835d5061133af195e3373355df5b_l3.png)
Comme
![Rendered by QuickLaTeX.com t - 1 \le 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-e24571ac5ca4109065a6e27c2c7a7851_l3.png)
![Rendered by QuickLaTeX.com 0 \le t \le 1](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ae45f9c56d1a8105bcbfebd78e2dcdab_l3.png)
![Rendered by QuickLaTeX.com \int_0^1 t^n \dfrac{t - 1}{ 1 + t^2} dt \le 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-144d72c4b77e7a3e799e8556e5b88823_l3.png)
![Rendered by QuickLaTeX.com I_{n + 1} - I_n \le 0](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-062919478421387f6887ada91b42e8fe_l3.png)
![Rendered by QuickLaTeX.com \left( I_n \right)_{n \in \mathbb{N}}](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-5e22b4c8673d9820351472c6d8d0077e_l3.png)
2) On encadre par deux suites ayant
pour limite.
![Rendered by QuickLaTeX.com 0\le t \le 1,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-acf07ffbdf316fff3e25d3e845a54768_l3.png)
![Rendered by QuickLaTeX.com 0 \le \dfrac{1}{1 + t^2} \le \dfrac12](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-2579effb19aa42596c40695712b31671_l3.png)
![Rendered by QuickLaTeX.com 0 \le \dfrac{t^n}{1 + t^2} \le \dfrac12 t^n](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-a34a88b1549b4cbd03e18dfa9bd0ae97_l3.png)
![Rendered by QuickLaTeX.com 0 \le I_n \le \dfrac12 \int_0^1 t^n = \dfrac{1}{2 \left( n + 1 \right)}.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-71acb6f7efb0adbc899ec068526d7f46_l3.png)
![Rendered by QuickLaTeX.com \lim_{n \to +\infty} \dfrac{1}{2 \left( n + 1 \right)} = 0,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-60a771b84bf887fe8065bb9fa0caefa8_l3.png)
![Rendered by QuickLaTeX.com \lim_{n \to +\infty} I_n = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-40461a1e02bf64fed8fdb19dbb326b77_l3.png)
3) On une intégration par parties en posant et
et
sont
sur
on a donc
![Rendered by QuickLaTeX.com I_n = \left[ \dfrac{1}{1 + t^2} \times \dfrac{t^{n + 1}}{n + 1} \right]_0^1 + \dfrac{2}{n + 1} \int_0^1 \dfrac{t^{n +2} }{1 + t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-90f40c382556de362040f72cf7a1a56b_l3.png)
![Rendered by QuickLaTeX.com = \dfrac{1}{2 \left( n + 1 \right)} + \dfrac{2}{n + 1} \int_0^1 \dfrac{t^{n +2} }{1 + t^2} dt](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-1c7aac76d289b2c06cf8a5715ae073bd_l3.png)
![Rendered by QuickLaTeX.com n I_n = \dfrac12 \times \dfrac{n}{n + 1} + \dfrac{2n}{n + 1} \int_0^1 \dfrac{t^{n + 2}}{1 + t^2} dt.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-bb9dc975d207f043bad56b8282cca34c_l3.png)
En procédant comme ci-dessus, on montre que
![Rendered by QuickLaTeX.com \lim_{n \to +\infty} \int_0^1 \dfrac{t^{n + 2}}{1 + t^2} dt = 0.](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-ab744a85da6ca37778d38f191b1f50aa_l3.png)
![Rendered by QuickLaTeX.com \lim_{n \to +\infty} \dfrac12 \times \dfrac{n}{n + 1} = \dfrac12,](https://groupe-reussite.fr/ressources/wp-content/ql-cache/quicklatex.com-caee472ad18fcbba75f0d10d2696155c_l3.png)