Derniers messages sur Zeste de Savoirhttps://zestedesavoir.com/forums/2019-03-02T10:27:58+01:00Les derniers messages parus sur le forum de Zeste de Savoir.Calcul d'une intégrale, message #1998012019-03-02T10:27:58+01:00florian6973/@florian6973https://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199801<p>Merci pour votre réponse ! J’ai essayé de calculer l’intégrale avec ce lien:
<a href="https://www.integral-calculator.com/">https://www.integral-calculator.com/</a>
<img src="/media/galleries/6557/563d6a8b-1d4e-41d9-b35d-05fd4ca58dd2.png" alt="image.png"></p>
<p>Il redonne la réponse du document, mais en précisant qu’il a supposé que (e-1)(e+1)>0, ce qui n’est pas le cas lorsque 0<e<1.
Cela laisserait penser, que dans le document, dans les calculs, l’auteur suppose cela aussi ; je n’ai de plus pas trouvé de formule concernant arctan(x * y).</p>
<p>J’aurais bien approfondi la seconde méthode, mais pour l’instant, je n’ai pas encore vu en cours les intégrales de contour et la formule de Cauchy.</p>Calcul d'une intégrale, message #1997842019-03-01T21:59:06+01:00Gawaboumga/@Gawaboumgahttps://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199784<p>Et bien, cela m’embête très fort. Il est noté que l’intégrale de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>θ</mi></msubsup><mfrac><mrow><mi>d</mi><mi>θ</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>e</mi><mi>cos</mi><mo></mo><mo>(</mo><mi>θ</mi><mo>)</mo><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\int_{0}^{\theta} \frac{d\theta}{(1 + e \cos(\theta))^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.564008em;vertical-align:-0.52em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.044008em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mathdefault mtight">e</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop mtight">cos</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> = <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mn>1</mn><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mn>2</mn></msup><mo>)</mo></mrow></mfrac><mo fence="false">[</mo><mfrac><mrow><mo>−</mo><mi>e</mi><mi>sin</mi><mo></mo><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>e</mi><mi>cos</mi><mo></mo><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mn>2</mn></msup></mrow></msqrt></mfrac><msup><mi>tan</mi><mo></mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>[</mo><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>e</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>e</mi></mrow></mfrac></msqrt><mi>tan</mi><mo></mo><mrow><mo>(</mo><mfrac><mi>θ</mi><mn>2</mn></mfrac><mo>)</mo></mrow><mo>]</mo><mo>+</mo><mi>n</mi><mn>2</mn><mi>π</mi><mo fence="false">]</mo></mrow><annotation encoding="application/x-tex">\frac{1}{(1 - e^2)} \Big[\frac{-e \sin(\theta)}{1 + e \cos(\theta)} + \frac{2}{\sqrt{1 - e^2}} \tan^{-1}[\sqrt{\frac{1 - e}{1 + e}} \tan{(\frac{\theta}{2})}] + n2\pi\Big]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mathdefault mtight">e</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop mtight">cos</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathdefault mtight">e</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop mtight">sin</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.8399999999999999em;vertical-align:-0.6341115em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.5445179999999996em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" 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Et pour <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi><mo>=</mo><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">\theta = 2\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span>, ça marche presque, au facteur <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>e</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>e</mi></mrow></mfrac></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\frac{1 - e}{1 + e}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8399999999999999em;vertical-align:-0.6341115em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2058885em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">+</span><span class="mord mathdefault mtight">e</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1658885em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width="400em" height="1.8800000000000001em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,
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M1001 80H400000v40H1013z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6341115em;"><span></span></span></span></span></span></span></span></span></span> dont on ne sait trop quoi faire. Peut-être qu’il existe une identité trigonométrique du type: <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>tan</mi><mo></mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>∗</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\tan^{-1} (x * y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0690879999999998em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">tan</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8190879999999999em;"><span style="top:-3.06798em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span> = ? et qu’on pourrait dire que ce terme ne contribue pratiquement pas (le cas <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">e = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span> me fait peur) mais j’en ai aucune idée.</p>
<p>Pour la seconde méthode, un changement de variable qui faisait intégrer de 1 à 1 et l’apparition d’une partie complexe (cercle) fait très fort penser à une intégrale de contour et à la formule de Cauchy, non ?</p>Calcul d'une intégrale, message #1997682019-03-01T11:17:25+01:00florian6973/@florian6973https://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199768<p>Bonjour,</p>
<p>Je remonte le fil, si quelqu’un a un conseil. Ce n’est pas grave sinon. Merci.</p>Calcul d'une intégrale, message #1994432019-02-24T09:48:02+01:00florian6973/@florian6973https://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199443<p>Merci pour votre lien ! C’est en effet ce que je cherchais. Cependant, je n’arrive pas à retrouver le résultat de la calculatrice lorsque je remplace <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span> par <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span> ou <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span> dans la formule du document. Ai-je raté quelque chose ?</p>
<p>Pour votre seconde méthode, si j’intègre entre 0 et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span>, le changement de variable me fait intégrer de 1 à 1. Est-ce tout simplement qu’il vaut mieux intégrer de 0 à <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></span> puis de multiplier par deux ?</p>
<p>Merci par avance pour votre réponse.</p>Calcul d'une intégrale, message #1994392019-02-23T21:41:42+01:00Gawaboumga/@Gawaboumgahttps://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199439<p>Et bien, c’est très <a href="https://www.whitman.edu/Documents/Academics/Mathematics/erickson.pdf">simple</a>: les calculs commencent page 26 et se terminent page 31.</p>
<p>Ou alors, on se dit qu’une ellipse, ça ressemble à un cercle. Et on regarde la forme exponentielle de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>cos</mi><mo></mo></mrow><annotation encoding="application/x-tex">\cos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mop">cos</span></span></span></span></span> et on pose <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">x = e^{i \theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></span></span></span></span></span></span>. C’est un poil moins calculatoire ;-)</p>Calcul d'une intégrale, message #1994282019-02-23T15:17:38+01:00florian6973/@florian6973https://zestedesavoir.com/forums/sujet/12146/calcul-dune-integrale/?page=1#p199428<p>Bonjour,</p>
<p>Je cherche actuellement à démontrer la troisième loi de Kepler. Pour cela, j’ai recours au changement de variable de Binet (<span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow><annotation encoding="application/x-tex">u=\frac{1}{r}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>). Cependant, je bloque sur un point. En effet, pour parvenir au résultat, je dois intégrer <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></msubsup><mfrac><mn>1</mn><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>e</mi><mo>⋅</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo>(</mo><mi>θ</mi><mo>)</mo><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mi mathvariant="normal">d</mi><mi>θ</mi></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \int_{0}^{2\pi} \frac{1}{(1+e \cdot cos(\theta))^2} \, \mathrm{d}\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5000080000000002em;vertical-align:-0.936em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5640080000000003em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathrm">d</span></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span></span>, avec le paramètre <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">e</span></span></span></span></span> strictement compris entre 0 et 1.</p>
<p>La calculatrice me donne un résultat qui permet bien de conclure (<span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>π</mi></mrow><mrow><mo>(</mo><msup><mi>e</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>)</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{-2\pi}{(e^2-1)\sqrt{1-e^2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.47559em;vertical-align:-0.630482em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.544518em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9221171428571429em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.882117142857143em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
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s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.11788285714285718em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">π</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.630482em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), mais j’aimerais savoir s’il est possible de la calculer soi-même. Pensez-vous que les règles de Bioche permettraient de conclure ? Je ne sais pas trop quel changement de variable poser.</p>
<p>Je vous remercie par avance pour votre réponse,</p>
<p>Bien cordialement,</p>
<p>florian6973.</p>