  This paper presents an analytical closed-form solution to improper integral
$\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is
based on splitting the improper integral into an infinite sum of definite
integrals with successive integer limits. The exact solution of every definite
integral is obtained by making use of the binomial polynomial expansion, which
then allows expression of the entire summation equivalently in terms of a
weighted sum of Riemann zeta functions. It turns out that the solution
fundamentally depends on whether or not $r$ is an integer. If $r$ is a
non-negative integer, then the solution is manifested in a finite series of
weighted Bernoulli numbers, which is then drastically simplified to a second
order rational function $\mu(r)=(-1)^{r+1}/(r+1)(r+2)$. This is achieved by
taking advantage of the relationships between Bernoulli numbers and binomial
coefficients. On the other hand, if $r$ is a non-integer real-valued number,
then we prove $\mu(r)=0$ by the virtue of the elegant relationships between
zeta and gamma functions and their properties.
