<p>Panel method, which models potential flow field by distributing sources and doublets over the body surfaces and wakes, remains a widely used approach in computational fluid dynamics for its simplicity and satisfactory accuracy. Evaluating influence coefficients of velocity potential and velocity induced by distributed source and doublet panels requires surface integrations over them. However, conventional formulations reported in the literature can exhibit singularity in certain cases, depending on the choice of the local coordinate system. Here, we derive a new set of analytical integration formulas that are inherently free from such singularities and possess clear, intuitive geometrical interpretations. In particular, the influence coefficient of velocity potential from doublet corresponds to the solid vertex angle of a pyramid subtended at the target point. The proposed formulations were implemented and validated through numerical examples, confirming their accuracy and robustness. These singularity-free expressions not only enhance physical understanding of the effects induced by sources and doublets but also make their implementation in panel codes straightforward and reliable.</p>