Voici une liste des formules importantes en analyse
0.1 Identités remarquables
\((a + b)^{2} = a^{2} + 2ab + b^{2}\)
\((a - b)^{2} = a^{2} - 2ab + b^{2}\)
\((a + b)(a - b) = a^{2} - b^{2}\)
\((a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}\)
\((a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}\)
\(a^{3} - b^{3} = (a - b)\left( a^{2} + ab + b^{2} \right)\)
\(a^{3} + b^{3} = (a + b)\left( a^{2} - ab + b^{2} \right)\)
1 Suites arithmétiques et géométriques
1.1 Suites arithmétiques
Termes de la suite (\(r\) désigne la raison):
\(u_{n+1} - u_{n} = r\)
\(u_{n} = u_{0} + nr\)
\(u_{n} = u_{p} + (n-p)r\)
Somme des termes :
\(\displaystyle S_{n} = \sum_{k=0}^{n}u_{k} = \dfrac{(n+1)(u_{0}+u_{n})}{2}\)
Cas général avec \(n_{1} \leq n_{2}\) :
\(\displaystyle S' = \sum_{k = n_{1}}^{n_{2}}u_{k} = \frac{(nombre\ de\ termes)(premier\ terme + dernier\ terme)}{2}\)
Cas particulier :
\(\displaystyle 1 + 2 + 3\ \cdots + (n - 1) + n = \frac{n(n + 1)}{2}\)
1.2 Suites géométriques
On suppose que la suite est non nulle. Termes de la suite ((q) désigne la raison) :
\(\dfrac{u_{n+1}}{u_{n}} = q\)
\(u_{n} = u_{0} \times q^{n}\)
\(u_{n} = u_{p}\times q^{n-p}\)
Somme des termes :
\(S_{n} = \sum\limits_{k=0}^{n}u_{k}\)
- Si \(q \neq 1\) : \(S_{n} = u_{0}\times \frac{1-q^{n+1}}{1-q}\)
- Si \(q=1\) : \(S_{n}=u_{0}(n+1)\)
Cas général avec \(n_{1} \leq n_{2}\) : \(S'_{n} = \sum\limits_{k=0}^{n}u_{k}\)
- Si \(q \neq 1\) : \(\displaystyle S'_{n} = (\text{premier terme}) \times \frac{1- \left( q^{\text{nombre de termes}} \right)}{1-q}\)
- Si \(q \neq 1\) : \(\displaystyle S'_{n} = (\text{premier terme}) \times \frac{1- q ^{(\text{nombre de termes})}}{1-q} = u_0 \times \frac{1-q^{n+1}}{1-q}\)
- Si \(q = 1\) : \(S'_{n} = (\text{nombre de termes}) \times (\text{premier terme})\)
2 Formules de dérivation
2.1 Formules générales
Dans ce qui suit, \(u\) et \(v\) désignent deus fonctions d’une variable réelle \(x\), et \(k\) une constante réelle.
\((u+v)' = u' + v'\)
\((ku)' = ku'\)
\((uv)' = u'v + uv'\)
\(\left( \dfrac{u}{v} \right)' = \dfrac{u'v - uv'}{v^{2}}\)
\((v\circ u)' = u' \times (v'\circ u)\)
\(\displaystyle(u^{-1})' = \dfrac{1}{u' \circ u^{-1}}\)
2.2 Fonctions usuelles
2.2.1 fonctions non composées
| Fonction \(f\) | \(\mathscr{D}_{f}\) | Fonction dérivée \(f'\) | \(\mathscr{D}_{f'}\) |
|---|---|---|---|
| \(k\) | \(\mathbb{R}\) | 0 | \(\mathbb{R}\) |
| \(x\) | \(\mathbb{R}\) | \(1\) | \(\mathbb{R}\) |
| \(\dfrac{1}{x}\) | \(\mathbb{R}^{*}\) | \(-\dfrac{1}{x^{2}}\) | \(\mathbb{R}^{*}\) |
| \(\sqrt{ x }\) | \(\mathbb{R}^{+}\) | \(\frac{1}{2\sqrt{ x }}\) | \(\mathbb{R}^{+*}\) |
| \(x^{n}\) avec \(n \in\mathbb{Z}\) | \(\mathbb{R}\) | \(nx^{n-1}\) | \(\mathbb{R} \text{ si } n \geq 0, \quad \mathbb{R}^{*} \text{ si } n < 0\) |
| \(x^{\alpha}\) avec \(\alpha \in\mathbb{R}\) | \(\mathbb{R}^{+} \text{ si } \alpha \geq 0, \quad \mathbb{R}^{+*} \text{ si } \alpha<0\) | \(\alpha x^{\alpha-1}\) | \(\mathbb{R} \text{ si } n \geq 0, \quad \mathbb{R}^{*} \text{ si } \alpha < 1\) |
| \(\ln \mid x \mid\) | \(\mathbb{R}^{+*}\) | \(\dfrac{1}{x}\) | \(\mathbb{R}^{+*}\) |
| \(\exp x\) | \(\mathbb{R}\) | \(\exp x\) | \(\mathbb{R}\) |
| \(\sin x\) | \(\mathbb{R}\) | \(\cos x\) | \(\mathbb{R}\) |
| \(\cos x\) | \(\mathbb{R}\) | \(-\sin x\) | \(\mathbb{R}\) |
| \(\tan x\) | \(\mathbb{R} \setminus \left\lbrace \dfrac{\pi}{2} + k\pi \right\rbrace\) | \(1+\tan^2 x = \dfrac{1}{\cos^2 x}\) | \(\mathbb{R} \setminus \left\lbrace \dfrac{\pi}{2} + k\pi \right\rbrace\) |
| \(\mathrm{sh} x\) | \(\mathbb{R}\) | \(\mathrm{ch} x\) | \(\mathbb{R}\) |
| \(\mathrm{ch} x\) | \(\mathbb{R}\) | \(\mathrm{sh} x\) | \(\mathbb{R}\) |
| \(\mathrm{th} x\) | \(\mathbb{R}\) | \(1-\mathrm{th}^2 x = \dfrac{1}{\mathrm{ch}^2 x}\) | \(\mathbb{R}\) |
Les dérivées des fonctions réciproques des fonctions trigonométriques et hyperboliques figurent dans la section “trigonométrie réciproque
2.2.2 Fonctions composées
| forme de la fonction | forme de la dérivée |
|---|---|
| \(\dfrac{1}{u}\) | \(-\dfrac{u'}{u^2}\) |
| \(\sqrt{u}\) | \(\dfrac{u'}{2 \sqrt{u}}\) |
| \(u^{\alpha}\) | \(\alpha u' u^{\alpha - 1}\) |
| $u $ | \(\dfrac{u'}{u}\) |
| \(\exp u\) | \(u' \times \exp u\) |
| \(\sin u\) | \(u'\times \cos u\) |
| \(\cos u\) | \(-u' \times \sin u\) |
| \(\tan u\) | \(u' \times (1+\tan^2 u) = \dfrac{u'}{\cos^2 u}\) |
| \(\mathrm{sh} u\) | \(u' \times \mathrm{ch} u\) |
| \(\mathrm{ch} u\) | \(u' \times \mathrm{sh} u\) |
| \(\mathrm{th} u\) | \(u' \times (1-\mathrm{th}^2 u) = \dfrac{u'}{\mathrm{ch}^2 u}\) |
3 Fonctions trigonométriques et hyperboliques
3.1 Fonctions trigonométriques
\(\cos^2 x + \sin^2 x = 1\)
\(\cos(a+b) = \cos a \cos b - \sin a \sin b\)
\(\cos(a-b) = \cos a \cos b + \sin a \sin b\)
\(\sin(a+b) = \sin a\cos b + \sin b \cos a\)
\(\sin(a-b) = \sin a \cos b - \sin b \cos a\)
\(\tan(a+b) = \dfrac{\tan a + \tan b}{1-\tan a \tan b}\)
\(\tan(a-b) = \dfrac{\tan a - \tan b}{1+\tan a \tan b}\)
\(\cos a \cos b = \dfrac{1}{2} \big( \cos(a+b)+\cos(a-b) \big)\)
\(\sin a\sin b = \dfrac{1}{2} \big( \cos(a-b) - \cos(a+b) \big)\)
\(\sin a \cos b = \dfrac12 \big( \sin(a+b) + \sin(a-b) \big)\)
\(\cos a + \cos b = 2\cos \dfrac{(a+b)}{2}\cos\dfrac{(a-b)}{2}\)
\(\cos a - \cos b = -2\sin\dfrac{a+b}{2}\sin\dfrac{a-b}{2}\)
\(\sin a + \sin b = 2\sin\dfrac{a+b}{2}\cos\dfrac{a-b}{2}\)
\(\sin a - \sin b = 2\sin\dfrac{a-b}{2} \cos\dfrac{a+b}{2}\)
\(\cos(2x) =\cos ^{2}x-\sin ^{2}x \quad= 2\cos ^{2}x-1 \quad= 1-2\sin ^{2}x \quad= \frac{1-\tan ^{2}x}{1+\tan ^{2}x}\)
\(\cos^2 x = \dfrac{1+\cos(2x)}{2}\)
\(\sin^2 x = \dfrac{1-\cos(2x)}{2}\)
\(\tan^2 x = \dfrac{1-\cos(2x)}{1+\cos(2x)}\)
\(\sin(2x) = 2\sin x\cos x \quad= 1+\tan ^{2}x\)
\(\tan(2x) = \dfrac{2\tan x}{1 - \tan^2 x}\)
3.2 Fonctions hyperboliques
\(\mathrm{ch}^2 x - \mathrm{sh}^2 x = 1\)
\(\mathrm{ch}(a+b) = \mathrm{ch} (a) \;\mathrm{ch} (a) + \mathrm{sh} a \mathrm{sh} b\)
\(\mathrm{ch}(a-b) = \mathrm{ch} (a)\; \mathrm{ch} (b) - \mathrm{sh} a \mathrm{sh} b\)
\(\mathrm{sh}(a+b) = \mathrm{sh} (a)\; \mathrm{ch} (b) + \mathrm{sh} (b)\; \mathrm{ch} (a)\)
\(\mathrm{sh}(a-b) = \mathrm{sh} (a); \mathrm{ch} (b) - \mathrm{sh} (b) \mathrm{ch} (a)\)
\(\mathrm{th}(a+b) = \dfrac{\mathrm{th} a + \mathrm{th} b}{1 + \mathrm{th} (a) \mathrm{th} (b)}\)
\(\mathrm{th}(a-b) = \dfrac{\mathrm{th} a - \mathrm{th} b}{1 - \mathrm{th} (a) \mathrm{th} (b)}\)
\(\mathrm{ch} (a) \;\mathrm{ch} (b) = \dfrac12 \big( \mathrm{ch}(a+b) + \mathrm{ch}(a-b) \big)\)
\(\mathrm{sh} (a)\; \mathrm{sh} (b) = \dfrac12 \big( \mathrm{ch}(a+b) - \mathrm{ch}(a-b) \big)\)
\(\mathrm{sh} (a)\; \mathrm{ch} (b) = \dfrac12 \big( \mathrm{sh}(a+b) + \mathrm{sh}(a-b) \big)\)
\(\mathrm{ch} (a) + \mathrm{ch} (b) = 2\mathrm{ch} \dfrac{a+b}{2} \mathrm{ch} \dfrac{a-b}{2}\)
\(\mathrm{ch} (a) - \mathrm{ch} (a) = 2\mathrm{sh} \dfrac{a+b}{2}\mathrm{sh}\dfrac{a-b}{2}\)
\(\mathrm{sh} (a) + \mathrm{sh} (b) = 2\mathrm{sh} \dfrac{a+b}{2}\mathrm{ch}\dfrac{a-b}{2}\)
\(\mathrm{sh} (a) - \mathrm{sh} (b) = 2\mathrm{sh}\dfrac{a-b}{2} \mathrm{ch}\dfrac{a+b}{2}\)
\(\mathrm{ch}(2x) = \mathrm{ch} ^{2}x + \mathrm{sh} ^{2}x \quad= 2\mathrm{ch} ^{2}x - 1 \quad= 1+2\mathrm{sh} ^{2}x \quad=\frac{1+\mathrm{th} ^{2}x}{1-\mathrm{th} ^{2}x}\)
\(\mathrm{ch}^2 x = \dfrac{1+\mathrm{ch}(2x)}{2}\)
\(\mathrm{sh}^2 x = \dfrac{\mathrm{ch}(2x) - 1}{2}\)
\(\mathrm{th}^2 x = \dfrac{\mathrm{ch}(2x) - 1}{\mathrm{ch}(2x) + 1}\)
\(\mathrm{sh}(2x) = 2\mathrm{sh}(x) \mathrm{ch}(x) \quad= \frac{2\mathrm{th}(x)}{1-\mathrm{th}^{2}(x)}\)
\(\mathrm{th}(2x) = \dfrac{2\mathrm{th} x}{1+\mathrm{th}^2 x}\)
3.3 Points sur le cercle trigonométrique
| \(-x\) | \(\frac{\pi}{2}+x\) | \(\frac{\pi}{2}-x\) | \(\pi+x\) | \(\pi-x\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | |
|---|---|---|---|---|---|---|---|---|---|---|
| \(\sin\) | \(-\sin x\) | \(\cos x\) | \(\cos x\) | \(-\sin x\) | \(\sin x\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{ 2 }}{2}\) | \(\frac{\sqrt{ 3 }}{2}\) | 1 |
| \(\cos\) | \(\cos x\) | \(-\sin x\) | \(\sin x\) | \(-\cos x\) | \(-\cos x\) | \(1\) | \(\frac{\sqrt{ 3 }}{2}\) | \(\frac{\sqrt{ 2 }}{2}\) | \(\frac{1}{2}\) | 0 |
| \(\tan\) | \(-\tan x\) | \(-\frac{1}{\tan x}\) | \(\frac{1}{\tan x}\) | \(\tan x\) | \(-\tan x\) | \(0\) | \(\frac{\sqrt{ 3 }}{2}\) | \(1\) | \(\sqrt{ 3 }\) |
3.4 Trigonométrie réciproque
\(\arcsin x + \arccos x = \dfrac{\pi}{2}\)
\(\arctan x + \arctan \dfrac{1}{x} = \text{sg}(x) \times \dfrac{\pi}{2}\) avec \(\mathrm{sg}(x) = 1 \text{ si } x>0\) et \(\mathrm{sg}(x) = -1 \text{ si } x < 0\)
\((\arcsin x)' = \dfrac{1}{\sqrt{1-x^2}}\)
\((\arccos x)' = -\dfrac{1}{\sqrt{1-x^2}}\)
\((\arctan x)' = \dfrac{1}{1+x^2}\)
\((\arcsin u)' = \dfrac{u'}{\sqrt{1-u^2}}\)
\((\arccos u)' = -\dfrac{u'}{\sqrt{1-u^2}}\)
\((\arctan u)' = \dfrac{u'}{1+u^2}\)
3.5 Trigonométrie hyperbolique réciproque
\(\arg\mathrm{sh} x = \ln \left( x + \sqrt{ 1+x^2 } \right)\)
\(\arg\mathrm{ch} x = \ln \left( x + \sqrt{ 1 - x^2 } \right)\)
\(\arg\mathrm{th} x = \dfrac12 \ln \left( \dfrac{1+x}{1-x} \right)\)
\((\arg\mathrm{sh} x)' = \dfrac{1}{\sqrt{ x^2 + 1 }}\)
\((\arg\mathrm{ch} x)' = \dfrac{1}{\sqrt{x^2 - 1}}\)
\((\arg\mathrm{th} x)' = \dfrac{1}{1-x^2}\)
\((\arg\mathrm{sh} u)' = \dfrac{u'}{\sqrt{ u^2 + 1 }}\)
\((\arg\mathrm{ch} u)' = \dfrac{u'}{\sqrt{ u^2 - 1}}\)
\((\arg\mathrm{th} u)' = \dfrac{u'}{1 - u^2}\)
4 Limites usuelles
4.1 Comportement à l’infini
\(\displaystyle\lim_{x\to+\infty} \ln x = +\infty\)
\(\displaystyle\lim_{x\to+\infty}\exp x = +\infty\)
\(\displaystyle\lim_{x\to-\infty}\exp x = 0\)
Si \(\alpha > 0\), \(\displaystyle\lim_{x\to+\infty}x^{\alpha} = +\infty\)
Si \(\alpha < 0\), \(\displaystyle\lim_{x\to+\infty}x^{\alpha} = 0\)
Si \(\alpha > 0\), \(\displaystyle\lim_{x\to+\infty}\dfrac{\exp x}{x^\alpha} = \lim_{x\to+\infty}\dfrac{e^x}{x^\alpha} = +\infty\)
Si \(\alpha > 0\), \(\displaystyle\lim_{x\to+\infty}x^\alpha\times\exp(-x) = \lim_{x\to+\infty}x^\alpha e^{-x} = 0\)
4.2 Comportement à l’origine
\(\displaystyle\lim_{x \rightarrow 0} \ln x = + \infty\)
Si \(\alpha > 0\), \(\displaystyle\lim_{x \rightarrow 0^+} x^\alpha = +\infty\)
Si \(\alpha < 0\), \(\displaystyle\lim_{x \rightarrow 0^+} x^\alpha = +\infty\)
Si \(\alpha > 0\), \(\displaystyle\lim_{x \rightarrow 0^+} \left( x^\alpha \ln x \right) = 0\) propriété de croissance comparée
\(\displaystyle \lim_{x \rightarrow 0} \dfrac{\ln (1+x)}{x} = 1\)
\(\displaystyle \lim_{x \to 0} \dfrac{e^x - 1}{x} = 1\)
\(\displaystyle \lim_{x \rightarrow 0} \dfrac{\sin x}x = 1\)
5 Formules d’intégration
5.1 Intégration par parties
\(\displaystyle \int_{a}^{b} f(x)g'(x) \, dx = \big[ f(x)g(x) \big]_{a}^{b} - \int_{a}^{b} f'(x)g(x) \, dx\)
5.2 Primitives usuelles
Si \(\alpha \neq -1\), \(\displaystyle \int x^\alpha \mathrm{d} x = \dfrac{1}{\alpha + 1}x^{\alpha +1} + \text{cte}\)
\(\displaystyle\int \dfrac{1}x \mathrm{d} x = \ln |x|+\text{cte}\)
\(\displaystyle\int \dfrac{1}{x+\alpha }\mathrm{d} x = \ln |x+\alpha |+\text{cte}\)
\(\displaystyle\int e^x \mathrm{d} x = e^x + \text{cte}\)
Si \(\alpha > 0\) et \(\alpha \neq 1\), \(\displaystyle\int \alpha^x \mathrm{d} x= \dfrac{1}{\ln \alpha }\times \alpha^x + \text{cte}\)
\(\displaystyle\int \cos x \mathrm{d} x = \sin x + \text{cte}\)
\(\displaystyle\int \sin x \mathrm{d} x = -\cos x + \text{cte}\)
Si \(\alpha \neq 0\), \(\displaystyle\int \cos \alpha x \mathrm{d} x = \dfrac{1}{\alpha} \sin \alpha x + \text{cte}\)
Si \(\alpha \neq 0\), \(\displaystyle\int \sin \alpha x \mathrm{d} x = \dfrac{1}{\alpha} \cos \alpha x + \text{cte}\)
\(\displaystyle \int \dfrac{1}{\cos ^2 x} \mathrm{d} x = \tan x + \text{cte}\)
\(\displaystyle\int 1 + \tan ^2 x \mathrm{d} x = \tan x + \text{cte}\)
\(\displaystyle\int \dfrac{1}{\sin ^2 x} \mathrm{d} x = -\mathrm{cotan} x + \text{cte} \quad= -\dfrac{1}{\tan x} + \text{cte}\)
\(\displaystyle\int \dfrac{1}{1+x^2} \mathrm{d} x = \arctan x + \text{cte}\)
\(\displaystyle\int \mathrm{ch} x \mathrm{d} x = \mathrm{sh} x + \text{cte}\)
\(\displaystyle\int \mathrm{sh} x \mathrm{d} x = \mathrm{ch} x + \text{cte}\)
\(\displaystyle\int \dfrac{1}{\mathrm{ch}^2 x} \mathrm{d} x = \mathrm{th} x + \text{cte}\)
\(\displaystyle\int \dfrac{1}{\mathrm{sh}^2 x} \mathrm{d} x = \dfrac{1}{\mathrm{th} x} + \text{cte}\)
5.3 Primitives de fonctions composées
\(\displaystyle\int \frac{u'(x)}{u(x)} \, dx = \ln(u) + \text{cte}\)