Finding the Area Under the Graph of \(f(x)=x\ln x\)

Find the area under the graph of \(f( x) =x\ln x\) from \(1\) to \(2.\)

Solution See Figure 1 for the graph of \(f( x) =x\ln x\). The area \(A\) under the graph of \(f\) from \(1\) to \(2\) is \( A=\int_{1}^{2}x\ln x\,dx.\) We use the integration by parts formula with \[ \begin{equation*} u=\ln x\qquad \hbox{and}\qquad dv=x\,dx \end{equation*} \]

Then \[ \begin{equation*} du=\frac{1}{x} dx\qquad \hbox{and}\qquad v=\int x dx=\frac{x^{2}}{2} \end{equation*} \]

and \[ \begin{eqnarray*} A=\int_{1}^{2}x\ln x\,dx&=&\left[ \frac{x^{2}}{2}\ln x\right] _{1}^{2}-\int_{1}^{2}\frac{x^{2}}{2}\left( \frac{1}{x} dx\right) =2\ln 2-\dfrac{1}{2}\int_{1}^{2}x\,dx\\[5pt] &=&2\ln 2- \dfrac{1}{2}\left[ \frac{x^{2}}{2}\right] _{1}^{2}=2\ln 2-\dfrac{3}{4} \end{eqnarray*} \]