Mathematics - Definite Integration Question with Solution | TestHub

$\underset{x\to \frac{\pi }{4}}{ \lim }\frac{{\displaystyle \frac{\mathrm{\pi }}{4}}{\int }_2^{se{c}^{2}x}f\left(x\right)dx}{x^2-{\displaystyle \frac{{\mathrm{\pi }}^2}{16}}}$

Using L-Hospital's rule, we get

 $=\underset{x\to \frac{\pi }{4}}{\lim }\frac{\frac{\pi }{4}f\left({\sec }^{2}x\right)·2·\sec x·\sec x·\tan x-0}{2x}$

$=\frac{\frac{\pi }{4}·2(\sqrt{2}{)}^2·(1)·f(2)}{2·\frac{\pi }{4}}$

$=2f\left(2\right)$