 Solution

With the inner product $⟨\chi ,\psi ⟩=\int {\chi }^{\ast }\psi dx$, the assumed Hermiticity of $\hat{x}$ and $\hat{p}$ givesConsequently the adjoint relation isAs with the stated Hermiticity assumption, these identities concern wavefunctions in the appropriate operator domains. Square integrability alone does not guarantee that derivatives or multiplication by $x$ are square integrable; one may first work with smooth rapidly decaying wavefunctions, where integration by parts has no boundary term, and then use the operator domains.