The transverse spin asymmetries measured in semi-inclusive leptoproduction of hadrons, when weighted with the hadron transverse momentum $P_T$, allow for the extraction of important transverse-momentum-dependent distribution functions. In particular, the weighted Sivers asymmetries provide direct information on the Sivers function, which is a leading-twist distribution that arises from a correlation between the transverse momentum of an unpolarised quark in a transversely polarised nucleon and the spin of the nucleon. Using the high-statistics data collected by the COMPASS Collaboration in 2010 with a transversely polarised proton target, we have evaluated two types of $P_T$-weighted Sivers asymmetries, which are both proportional to the product of the firsttransverse moment of the Sivers function and of the fragmentation function. The results are compared to the standard unweighted Sivers asymmetries and used to extract the first transverse moments of the Sivers distributions for $u$ and $d$ quarks.
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