This Quantum World/Appendix/Indefinite integral

The indefinite integralEdit

How do we add up infinitely many infinitesimal areas? This is elementary if we know a function   of which   is the first derivative. If   then   and


All we have to do is to add up the infinitesimal amounts   by which   increases as   increases from   to   and this is simply the difference between   and  

A function   of which   is the first derivative is called an integral or antiderivative of   Because the integral of   is determined only up to a constant, it is also known as indefinite integral of   Note that wherever   is negative, the area between its graph and the   axis counts as negative.

How do we calculate the integral   if we don't know any antiderivative of the integrand  ? Generally we look up a table of integrals. Doing it ourselves calls for a significant amount of skill. As an illustration, let us do the Gaussian integral


For this integral someone has discovered the following trick. (The trouble is that different integrals generally require different tricks.) Start with the square of  :


This is an integral over the   plane. Instead of dividing this plane into infinitesimal rectangles   we may divide it into concentric rings of radius   and infinitesimal width   Since the area of such a ring is   we have that


Now there is only one integration to be done. Next we make use of the fact that   hence   and we introduce the variable  :


Since we know that the antiderivative of   is   we also know that


Therefore   and


Believe it or not, a significant fraction of the literature in theoretical physics concerns variations and elaborations of this basic Gaussian integral.

One variation is obtained by substituting   for  :


Another variation is obtained by thinking of both sides of this equation as functions of   and differentiating them with respect to   The result is