e be the eeccenticity of a hyperbola and f(e) be the eccerticity of its conjugate hyperboal, and (fofofo....... of n times) (e) is F(e) Then ${\int }_1^3 \mathrm{F}\left(\mathrm{e}\right) \mathrm{de}$

If e and e₁ are eccentricities of a hyperbola and its conjugate then $\frac{1}{{\mathrm{e}}^2}+\frac{1}{{\mathrm{e}}_1^2}=1\Rightarrow {\mathrm{e}}_1=\frac{\mathrm{e}}{\sqrt{{\mathrm{e}}^2-1}}$

$\therefore \mathrm{f}\left(\mathrm{e}\right)=\frac{\mathrm{e}}{\sqrt{{\mathrm{e}}^2-1}},\left(\mathrm{fof}\right)\left(\mathrm{e}\right)=\mathrm{e},\text{ We therefore }$

$\underset{\mathrm{n}\text{ times }}{\underbrace{\text{ fofof ...of }\left(\mathrm{e}\right)}}=\left\{\begin{array}{l}\mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{n}\text{ is even }\\ \frac{\mathrm{e}}{\sqrt{{\mathrm{e}}^2-1}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{n}\text{ is odd }\end{array}\right.$