Using the definition of hermitian operator $A$: $$\int \psi^*(A\phi)dV=\int(A\psi)^*\phi dV,$$ together with the orbital angular momentum $$L_j=~\varepsilon_{jk\ell}x_k p_\ell=-i\hbar~\varepsilon_{jk\ell} x_k\partial_\ell,$$ to show that $L_j$ is hermitian. Here, I've used Einstein's summation convention for repeated indices.
Now, I am stuck at integration by parts. The LHS is $$-i\hbar\varepsilon_{jk\ell}\int\psi^*x_k(\partial_\ell\phi)dV\\ =-i\hbar\varepsilon_{jk\ell}\psi^*x_k\int (\partial_\ell\phi)dV +i\hbar\varepsilon_{jk\ell}\int\left[ \partial_\ell(\psi^*x_k)\int (\partial_\ell\phi)dV\right]dV\\ =-i\hbar\varepsilon_{jk\ell}\psi^*x_k\int (\partial_\ell\phi)dV +i\hbar\varepsilon_{jk\ell}\int\left[ \{(\partial_\ell\psi^*)x_k+\psi^*\delta_{k\ell}\}\int (\partial_\ell\phi)dV\right]dV\\ =-i\hbar\varepsilon_{jk\ell}\psi^*x_k\int (\partial_\ell\phi)dV +i\hbar\varepsilon_{jk\ell}\int\left[(\partial_\ell\psi^*)x_k\int (\partial_\ell\phi)dV\right]dV $$ where I have difficulty writing the next step. Am I doing it right? Any hint on how to proceed after this?
In your calculation, I sadly cant see how you can pull out some things out of the integral, but here is a solution: \begin{align} \int \psi_j^* L_j \phi_j dV&=\int \psi_j^* (-i \epsilon_{jkl}x_k \partial_l\phi_j)dV \\&=\int -i \epsilon_{jkl} x_k \psi_j^* \partial_l \phi_j dV \\ & =-\int \epsilon_{jkl} \partial_l (-ix_k \psi_j^*) \phi_j dV \\ &=-\int \epsilon_{jkl}(-i \delta_{kl}\psi_j^*-ix_k \partial_l \psi_j^*)\phi_j dV \\ &=-\int (-i \underbrace{\epsilon_{jkl} \delta_{kl}}_{=0}\psi_j^*-i \epsilon_{jkl} x_k \partial_l \psi^*)\phi_j dV \\ &=-\int (-i\epsilon_{jkl}x_k \partial_l \psi^*_j)\phi_j dV \\ &=\int (-i\epsilon_{jkl}x_k \partial_l \psi_j)^* \phi_j dV \\ &=\int (L_j \psi_j)^* \phi_j dV \end{align} Hope I didnt mess up any index, I assumed that all boundary terms vanish, which is a reasonable assumption for dense domains in $L^2(\mathbb{R}^3)$.