Example 7. Find the length of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.
Solution. We will use the parametric equations of the ellipse and the corresponding formula (4) for the length: $x=a \cos t, y=b \sin t$, $d x=-a \sin t d t, d y=b \cos t d t, \sqrt{d x^{2}+d y^{2}}=\sqrt{a^{2} \sin ^{2} t+b^{2} \cos ^{2} t} d t=$ $=\sqrt{\left(b^{2}-a^{2}\right) \cos ^{2} t+a^{2}} d t=\sqrt{a^{2}+c^{2} \cos ^{2} t} d t=a \sqrt{1+\varepsilon^{2} \cos ^{2} t} d t$, where $c^{2}=b^{2}-a^{2}, \varepsilon=\frac{c}{a}$ - the eccentricity of the ellipse.

Therefore, $L=a \int_{0}^{2 \pi} \sqrt{1+\varepsilon^{2} \cos ^{2} t} d t$

This is one of the "non-integrable" integrals, i.e., it cannot be expressed in terms of elementary functions. The corresponding non-elementary function, depending on $\varepsilon$, is called elliptic. It is tabulated and is used in the theory of functions of a complex variable, approximation theory, and other areas of mathematics.