Integration: Antiderivatives
Antiderivatives of some known functions
We have seen the antiderivatives of power functions. We will now discuss the antiderivatives of some other known functions.
\[\int \frac{1}{x} \;\dd x= \ln |x| + \green C \]
Example
#\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}#
\[\int \e^x \;\dd x=\e^x + \green C \]
Example
#\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}#
\[\int \blue a^x \;\dd x= \frac{\blue a^x}{\ln(\blue a)} + \green C \]
Example
#\begin{array}{rcl}
\displaystyle \int \blue {3}^ x \;\dd x&=& \dfrac{\blue 3^x}{\ln(\blue 3)} + \green C
\end{array}#
\[\int \ln(x) \; \dd x= {x \ln (x) -x }+ \green C \]
Example
#\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}#
\[\int \log_\blue {a}(x) \; \dd x= \frac{x \ln (x) -x }{\ln (\blue a) }+ \green C \]
Example
#\begin{array}{rcl}
\displaystyle \int \log_\blue {7}(x)\;\dd x &=& \dfrac{x \ln (x) -x }{\ln (\blue 7) }+ \green C
\end{array}#
\[\int \sin(x) \; \dd x = -\cos(x) + \green C \]
Example
#\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}#
\[\int \cos(x) \; \dd x= \sin(x) + \green C \]
Example
#\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)=# #2\cdot \e^x#
\[\begin{array}{rcl}
\displaystyle \int f(x) \, \dd x&=&\displaystyle \int 2\cdot \e^x \, \dd x\\
&&\phantom{xxx}\blue{\text{substituted}}\\
&=&\displaystyle 2\cdot \int \e^x \, \dd x\\
&&\displaystyle\phantom{xxx}\blue{\text{rule of calculation }\int c \cdot f(x) \; \dd x=c\int f(x) \; \dd x}\\
&=&\displaystyle2\cdot \e^x+C\\
&&\displaystyle\displaystyle\phantom{xxx}\blue{\text{rule of calculation }\int \e^x \, \dd x = \e^x + C}
\end{array}\]
Since only one antiderivative is asked, we can choose #C=0#. This gives
\[F(x)=2\cdot \e^x\]
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