Intégration: Primitives
Primitives de fonctions usuelles
Nous avons vu les primitives de fonctions puissance. Nous allons maintenant voir les primitives d'autres fonctions usuelles.
\[\int \frac{1}{x} \;\dd x= \ln |x| + \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle \int \frac{6}{x} \;\dd x&=&6\cdot \displaystyle \int \frac{1}{x} \;\dd x \\
&=& 6 \cdot \ln |x| + \green C
\end {array} #
\[\int \e^x \;\dd x=\e^x + \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle \int 3 \cdot \e^ x \;\dd x&=& 3 \displaystyle \int \e^x \; \dd x \\&=& 3 \cdot \e^x + \green C
\end {array} #
\[\int \blue a^x \;\dd x= \frac{\blue a^x}{\ln(\blue a)} + \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle \int \blue {3}^ x \;\dd x&=& \dfrac{\blue 3^x}{\ln(\blue 3)} + \green C
\end {array} #
\[\int \ln(x) \; \dd x= {x \ln (x) -x }+ \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle \int 4 \ln(x)\;\dd x &=& 4 \displaystyle \int \ln(x) \; \dd x \\ &=& 4 x \ln (x) - 4 x + \green C
\end {array} #
\[\int \log_\blue {a}(x) \; \dd x= \frac{x \ln (x) -x }{\ln (\blue a) }+ \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle \int \log_\blue {7}(x)\;\dd x &=& \dfrac{x \ln (x) -x }{\ln (\blue 7) }+ \green C
\end {array} #
\[\int \sin(x) \; \dd x = -\cos(x) + \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle\int 5 \cdot \sin(x) \; \dd x &=&\displaystyle5 \cdot \int \sin(x) \; \dd x \\
&=&\displaystyle -5 \cdot \cos(x) + \green C
\end {array} #
\[\int \cos(x) \; \dd x= \sin(x) + \green C \]
Exemple
# \begin{array}{rcl}
\displaystyle\int 3 \cdot \cos(x) \; \dd x &=&\displaystyle3 \cdot \int \cos(x) \; \dd x \\
&=&\displaystyle 3 \cdot \sin(x) + \green C
\end {array} #
#F(x)=# #4\cdot \e^x#
#\begin{array}{rcl}
\displaystyle \int f(x) \, \dd x&=&\displaystyle \int 4\cdot \e^x \, \dd x\\
&&\phantom{xxx}\blue{\text{substitution}}\\
&=&\displaystyle 4\cdot \int \e^x \, \dd x\\
&&\displaystyle\phantom{xxx}\blue{\text{règle de calcul }\int c \cdot f(x) \; \dd x=c\int f(x) \; \dd x}\\
&=&\displaystyle4\cdot \e^x+C\\
&&\displaystyle\displaystyle\phantom{xxx}\blue{\text{règle de calcul }\int \e^x \, \dd x = \e^x + C}
\end{array}#
Comme une primitive est demandée, nous pouvons choisir #C=0#. Donc:
\[F(x)=4\cdot \e^x\]
Ou visitez omptest.org si jou prenez un examen de OMPT.
