This Quantum World/Appendix/Indefinite integral

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The indefinite integral[edit]

How do we add up infinitely many infinitesimal areas? This is elementary if we know a function F(x) of which f(x) is the first derivative. If f(x)=\frac{dF}{dx} then dF(x)=f(x)\,dx and

\int_a^b f(x)\,dx=\int_a^b dF(x)=F(b)-F(a).

All we have to do is to add up the infinitesimal amounts dF by which F(x) increases as x increases from a to b, and this is simply the difference between F(b) and F(a).

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

How do we calculate the integral I=\int_a^b dx\,f(x) if we don't know any antiderivative of the integrand f(x)? 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 I:


This is an integral over the x{-}y plane. Instead of dividing this plane into infinitesimal rectangles dx\,dy, we may divide it into concentric rings of radius r and infinitesimal width dr. Since the area of such a ring is 2\pi r\,dr, we have that


Now there is only one integration to be done. Next we make use of the fact that \frac{d\,r^2}{dr}=2r, hence dr\,r=d(r^2/2), and we introduce the variable w=r^2/2:


Since we know that the antiderivative of e^{-w} is -e^{-w}, we also know that


Therefore I^2=2\pi 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 \sqrt{a}\,x for x:


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