\[\int \frac{1}{x} \;\dd x= \ln |x| + \green C \]

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}#