Problems on Techniques of Integration

Use the Integration by Parts. Set

\begin{displaymath}\left\{\begin{array}{lll}
u &=& \ln(x)\\
dv &=& dx\;.
\end{array}\right.\end{displaymath}

Then

\begin{displaymath}\left\{\begin{array}{lll}
du &=&\displaystyle \frac{1}{x} dx\\
v &=& x\;.
\end{array}\right.\end{displaymath}

So

\begin{displaymath}\int \ln(x) dx = x \ln(x) - \int x \frac{1}{x} dx = x \ln(x) - \int dx\end{displaymath}

or

\begin{displaymath}\int \ln(x) dx = x \ln(x) - x + C\;.\end{displaymath}

Detailed Answer.


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