It would be easier to generalize this problem

Lets do it in a generalised manner,

$f(y) = \int_{0}^{\pi/2} ln( y^{2}cos^{2}x + sin^{2}x)$

Here- $\frac{2468}{990} = y , \theta = x$

$f'(y) = 2y \int_{0}^{\pi/2}\frac{cos^{2}x}{sin^{2}x + y^{2}cos^{2}x}dx$

$= 2y \int_{0}^{\pi/2}\frac{dx}{tan^{2}x + y^{2}}$

$= 2y \int_{0}^{\pi/2}\frac{sec^{2}x - tan^{2}x }{tan^{2}x + y^{2}}dx$

$= 2y . \frac{1}{y} tan^{-1}( \frac{1}{y}) |_{0}^{\infty} -2y\frac{\pi}{2} + y^{2}f'(y)$

$f'(y) = \frac{\pi}{1 + y}$

$f(y) = \pi ln(1 + y) + c$

$y = 1 , f(1) = 0 , c = -\pi ln2$

$f(y) = \pi log(1 + y) -\pi ln2$

https://brilliant.org/profile/sandeep-125d73/sets/integrals/188242/problem/ans-is-my-favourite-no/ very easy to guess the ans :p