Simple applications of 'differentiation under intergal sign'
Standard integrals are sets of integrals which we encounter frequently in physics. This is mainly due to the universal nature of the functions involved in such integrals which turn out to be related somehow to the basic equations of the branch of physics under study. For example, while studying atomic and molecular physics we repeatedly encounter exponential functions because in general the atomic wave-functions contain exponentials. Anything things we do with these wave-functions like trying to find out transition probabilities or working out molecular orbitals, etc. we come across integrals of exponentials time and again. It is not convenient to evaluate these integrals every single time. So we try to prepare a set of integrals in terms of which we can usually express more complicated integrals that we encounter while solving problems. Similarly, in electrostatics we always try to find electric potential by solving Laplace equation for a given system and then using this po...