# Formule

tabel de primitive nedefinite

Mulțimea primitivelor
$\displaystyle \int u^n(x)u'(x) dx =$ $\displaystyle \frac{u^{n+1}(x)}{n+1} +C$
$n \in \mathbb{N}$
$\displaystyle \int u^a(x)u'(x) dx =$ $\displaystyle \frac{u^{a+1}(x)}{a+1} +C$
$a \in \mathbb{R} \setminus \{-1\}$, $u(I) \subset (0, \infty)$
$\displaystyle \int \frac{u'(x)}{u(x)} dx =$ $\displaystyle \ln{ \mid u(x) \mid } +C$
$u(x) \neq 0$, $\forall x \in I$
$\displaystyle \int a^{u(x)} u'(x) dx =$ $\displaystyle \frac{a^{u(x)}}{ \ln {a}} +C$
$a \in \mathbb{R}^{\ast}_{+} \setminus \{1\}$
$\displaystyle \int \frac{u'(x)}{u^2(x) + a^2} dx =$ $\displaystyle \frac{1}{a}\text{arctg}{ \frac{u(x)}{a} } +C$
$a \ne 0$
$\displaystyle \int \frac{u'(x)}{u^2(x) - a^2} dx =$ $\displaystyle \frac{1}{2a}\ln{ \left| \frac{u(x)-a}{u(x)+a} \right| } +C$
$u(x) \ne \pm a$, $\forall x \in I$, $a \ne 0$
$\displaystyle \int \sin{u(x)} u'(x) dx =$ $\displaystyle -\cos{u(x)} +C$
$\displaystyle \int \cos{u(x)} u'(x) dx =$ $\displaystyle \sin{u(x)} +C$
$\displaystyle \int \frac{u'(x)}{\cos^2{u(x)}} dx =$ $\displaystyle \text{tg}{u(x)} +C$
$u(x) \notin \displaystyle \left\{ \frac{ (2k+1) \pi}{2} | k \in \mathbb{Z} \right\}$, $\forall x \in I$
$\displaystyle \int \frac{u'(x)}{\sin^2{u(x)}} dx =$ $\displaystyle -\text{ctg}{u(x)} +C$
$u(x) \notin \displaystyle \left\{ k \pi | k \in \mathbb{Z} \right\}$, $\forall x \in I$
$\displaystyle \int \text{tg}{(u(x))} u'(x) dx =$ $\displaystyle -\ln{\left| \cos{u(x)} \right|} +C$
$u(x) \notin \displaystyle \left\{ \frac{ (2k+1) \pi}{2} | k \in \mathbb{Z} \right\}$, $\forall x \in I$
$\displaystyle \int \text{ctg}{(u(x))} u'(x) dx =$ $\displaystyle \ln{\left| \sin{u(x)} \right|} +C$
$u(x) \notin \displaystyle \left\{ k \pi | k \in \mathbb{Z} \right\}$, $\forall x \in I$
$\displaystyle \int \frac{u'(x)}{ \sqrt{ u^2(x) + a^2} } dx =$ $\displaystyle \ln{\left( u(x) + \sqrt{u^2(x)+a^2} \right)} +C$
$a \ne 0$
$\displaystyle \int \frac{u'(x)}{ \sqrt{ u^2(x) - a^2 } } dx =$ $\displaystyle \ln{\left| u(x) + \sqrt{u^2(x)-a^2} \right|} +C$
$a \gt 0$ și $u(I) \subset (-\infty , -a)$ sau
$a \gt 0$ și $u(I) \subset (a, \infty)$
$\displaystyle \int \frac{u'(x)}{ \sqrt{ a^2 - u^2(x) } } dx =$ $\displaystyle \arcsin{ \frac{u(x)}{a} } +C$
$a \gt 0$ și $u(I) \subset (-a, a)$