Title: [PTSI] Primitives

fonction: \( x \mapsto x^n, \quad n \in \mathbb{N}\)
primitive: \( x \mapsto \frac{x^{n+1}}{n+1}\)
domaine: \( \mathbb{R} \)

fonction: \( x \mapsto \frac{1}{x^{n}}, \quad n \in \mathbb{N}, n \geqslant 2\)
primitive: \( x \mapsto \frac{-1}{(n-1)x^{n-1}}\)
domaine: \( \mathbb{R}_+^* \text{ ou }\mathbb{R}_-^* \)

fonction: \( x \mapsto x^a, \quad a \in \mathbb{R} \setminus \mathbb{Z}\)
primitive: \( x \mapsto \frac{x^{a+1}}{a+1}\)
domaine: \(\mathbb{R_+^*}\)

fonction: \( x \mapsto \frac{1}{x}\)
primitive: \( x \mapsto \ln{(\vert x \vert)}\)
domaine: \(\mathbb{R}_+^* \text{ ou }\mathbb{R}_-^*\)

fonction: \( \exp  \)
primitive: \(\exp\)
domaine: \( \mathbb{R} \)

fonction: \(\cos \)
primitive: \(\sin \)
domaine: \( \mathbb{R} \)

fonction: \( \sin \)
primitive: \( - \cos \)
domaine: \( \mathbb{R} \)

fonction: \(x \mapsto 1 + \tan(x)^2 \)
primitive: \( \tan \)
domaine: \( \left] - \frac{\pi}{2}+ k \pi; \frac{\pi}{2} + k\pi \right[ ~~(k \in \mathbb{Z}) \)

fonction: \( x \mapsto \frac{1}{\cos(x)^2} \)
primitive: \( \tan\)
domaine: \( \left] - \frac{\pi}{2}+ k \pi; \frac{\pi}{2} + k\pi \right[ ~~(k \in \mathbb{Z}) \)

fonction: \( \tan\)
primitive: \( x \mapsto - \ln(\vert \cos(x) \vert ) \)
domaine: \( \left] - \frac{\pi}{2}+ k \pi; \frac{\pi}{2} + k\pi \right[ ~~(k \in \mathbb{Z}) \)

fonction: \( \operatorname{ch} \)
primitive: \( \operatorname{sh} \)
domaine: \( \mathbb{R} \)

fonction: \( \operatorname{sh} \)
primitive: \( \operatorname{ch}\)
domaine: \( \mathbb{R} \)

fonction: \( x \mapsto \frac{1}{1 + x^2} \)
primitive: \(\operatorname{Arctan} \)
domaine: \( \mathbb{R} \)

fonction: \( x \mapsto \frac{1}{\sqrt{1 - x^2}}\)
primitive: \( \operatorname{Arcsin}\)
domaine: \( ]-1;1[ \)

fonction: \( x \mapsto \frac{-1}{\sqrt{1 - x^2}}\)
primitive: \( \operatorname{Arcos} \)
domaine: \( ]-1;1[ \)

fonction: \( x \mapsto u'(x) u(x)^a, \quad a \in \mathbb{R} \setminus \{ -1 \}\)
primitive: \(x \mapsto \frac{u(x)^{a+1}}{a+1} \)
domaine: \( I \)

fonction: \( x \mapsto \frac{u'(x)}{u(x)} \)
primitive: \( x \mapsto \ln(\vert u(x) \vert)\)
domaine: \( I \)

fonction: \( x \mapsto u'(x) \operatorname{e}^{u(x)}\)
primitive: \( x \mapsto  \operatorname{e}^{u(x)}\)
domaine: \( I \)

fonction: \( x \mapsto - \sin(u(x))u'(x) \)
primitive: \( x \mapsto \cos(u(x))\)
domaine: \( I \)

fonction: \( x \mapsto \cos(u(x))u'(x)\)
primitive: \( x \mapsto \sin(u(x))\)
domaine: \( I \)

fonction: \( \ln \)
primitive: \( x \mapsto x \ln(x) - x \)
domaine: \( ]0,+\infty[ \)

fonction: \( x \mapsto \lambda, \quad \lambda \in \mathbb{R}\)
primitive: \( x \mapsto \lambda x \)
domaine: \( \mathbb{R} \)

fonction: \( x \mapsto a^x, \quad a \in \mathbb{R}_+^*,~ a \neq 1\)
primitive: \( x \mapsto \frac{1}{\ln(a)} a^x \)
domaine: \( \mathbb{R} \)

fonction: \( x \mapsto u'(x) \ln(u(x)) \)
primitive: \( x \mapsto u(x) \ln(u(x)) - u(x)\)
domaine: \( I \)

fonction: \( x \mapsto \operatorname{ch} (u(x))u'(x)\)
primitive: \( x \mapsto \operatorname{sh} (u(x))\)
domaine: \( I \)

fonction: \( x \mapsto \operatorname{sh} (u(x))u'(x)\)
primitive: \( x \mapsto \operatorname{ch} (u(x))\)
domaine: \( I \)

fonction: \( x \mapsto \frac{u'(x)}{ 1 + u^2(x)} \)
primitive: \( x \mapsto \operatorname{Arctan} (u(x))\)
domaine: \( I \)

fonction: \( x \mapsto \tan(u(x))u'(x)\)
primitive: \( x \mapsto -\ln(\vert \cos ( u(x) ) \vert ) \)
domaine: \( I \)