Problems on Techniques of Integration

Problem. Evaluate

\begin{displaymath}\int x^n \ln(x) dx\;\;\;\mbox{where}\;\;n = 1,2,\cdots\end{displaymath}

Short Answer. Use the Integration by Parts. Set

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

Then

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

So

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

or

\begin{displaymath}\int x^n \ln(x) dx = \frac{x^{n+1}}{n+1} \ln(x) - \frac{x^{n+1}}{(n+1)^2} + C\;.\end{displaymath}

Detailed Answer.


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