Assume that $x(t)$ is a white Gaussian process with PSD $\sigma^2$; as a result, $$\mathbb{E}[x(t)x(t-\tau)]=\sigma^2\delta(\tau).$$ Define the random variable $X$ as $$X=\int_{0}^{T}x^2(t)~\text{d}t,$$ where $T\in\mathbb{R}^+$ is finite. What is the distribution of $X$?
P.S. I tried to treat the integral as Riemann summation, to have a summation of Chi-squared random variables. However, as the variance of the normal random variable at each time instant is infinit, I didn't have a progress.