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This (in particular the 2nd equality) is incorrect. Let $\omega \in \Omega$ denote the sample realization. You state:
$$\int_{0}^{T} B_{t} (\omega) d t=T^{1+1/2} \int_{0}^{1} B_{s}(\omega) d s$$
For example assume $T>1$. This is claiming that in order to compute the integral you only need to consider realizations of $B_t$ between $[0,1]$ and then scale ...
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