Example 5. Find the length of the arc of the curve $y=\frac{x^{2}}{4}-\frac{\ln x}{2}$, enclosed between the points with abscissas $x=1$ and $x=e$.
Solution. We use formula (3). We have

$$
y^{\prime}=\frac{x}{2}-\frac{1}{2 x}=\frac{x^{2}-1}{2 x}, 1+y^{\prime 2}=1+\left(\frac{x^{2}-1}{2 x}\right)^{2}=\frac{\left(1+x^{2}\right)^{2}}{4 x^{2}}
$$

and

$$
\begin{aligned}
& L=\int_{1}^{e} \sqrt{1+y^{\prime 2}} d x=\int_{1}^{e} \frac{1+x^{2}}{2 x} d x=\left.\frac{1}{2}\left(\ln x+\frac{x^{2}}{2}\right)\right|_{1} ^{e}= \\
&=\frac{1}{2}\left(1+\frac{e^{2}-1}{2}\right)=\frac{e^{2}+1}{4}
\end{aligned}
$$

Answer. $L=\frac{1}{4}\left(e^{2}+1\right)$.