I present here a simple approximate expression for $\zeta{(3)}$, the
Ap\'ery's constant, which is accurate to 21 digits. This closed-form expression
has been found experimentally via the PSLQ algorithm, with a search basis
composed by some suitable real numbers involving $ \pi$, $ \ln{2} $, $
\ln{(1+\sqrt{2})}$, and $G$ (the Catalan's constant). The very simple
\emph{Maple} code written for finding the rational coefficients of this
expression is also shown.

In this short paper, I introduce a simple method for exactly evaluating the
definite integrals $\int_0^{\pi}{\ln{(\sin{\theta})} d\theta}$,
$\int_0^{\pi/2}{\ln{(\sin{\theta})} d\theta}$,
$\int_0^{\pi/2}{\ln{(\cos{\theta})} d\theta}$, and
$\int_0^{\pi/2}{\ln{(\tan{\theta})} d\theta}$ in finite terms. The method
consists in taking into account certain products of trigonometric functions at
rational multiples of $\pi$ whose logarithm yields sums that can be promptly
written in the form of Riemann sums, and then to take the limit as the number
of terms tends to infinity.