I've been asked with computing the $L^2$ variation of $\int_0^tW_sds$, where $W_t$ is a Brownian motion. I am not supposed to use to stochastic calculus to solve, but the definition of quadratic variation and perhaps the fact that for a Brownian motion $W_t$ the quadratic variation in $L^2$ is $t$ (the length of the interval of the index set).
Is there a slick way to use this fact to calculate the quadratic variation? So far I have gone by brute force to show that quadratic variation of the integral in $L^2$ is $0$ (which is quite lengthy so I have not yet written them).
I will appreciate any advice, or at least an indication whether $0$ is indeed the $L^2$ quadratic variation.