Flip Into a New Dimension and Solve the Integral!
Generally speaking, if the antiderivative is easy to find, the Newton-Leibniz formula is the universal method for definite integrals. But the Newton-Leibniz formula only applies to integrals of continuous functions—if the interval of integration contains a singularity, it no longer holds. For example, consider the integral
$$\int_{-1}^1 \frac{1}{x^2}dx$$
Of course, strictly speaking from a mathematical standpoint, this expression doesn't even make sense, since the integrand is undefined at the origin. Still, from a physical point of view, by symmetry we're confident that
$$\int_{-1}^1 \frac{1}{x^2}dx=2\int_{0}^1 \frac{1}{x^2}dx=\lim_{\varepsilon\to 0}2\int_{\varepsilon}^1 \frac{1}{x^2}dx$$
and thus conclude that the integral diverges. This treatment is acceptable to some extent, but it isn't entirely satisfying, because it forces us to split the integral into pieces. Is there some way to handle this situation directly? Indeed there is: again introduce a parameter, and eventually let that parameter go to zero. Consider the parametrized integral
$$\int_{-1}^1 \frac{1}{x^2+\varepsilon^2}dx$$
As long as the parameter is positive, this integrand is continuous everywhere on $\mathbb{R}$—that is, the singularity disappears—so we can apply the Newton-Leibniz formula
$$\int_{-1}^1 \frac{1}{x^2+\varepsilon^2}dx=\left.\frac{1}{\varepsilon}\arctan\left(\frac{x}{\varepsilon}\right)\right|_{-1}^{1}$$
Taking the limit $\varepsilon\to 0$, we automatically arrive at the conclusion that the integral diverges. more
However, consider the integral
$$\int_{-1}^1 \frac{1}{x}dx$$
Here the above trick no longer works, because after introducing the parameter
$$\int_{-1}^1 \frac{1}{x+\varepsilon}dx$$
the singularity is still sitting inside the region of integration—it has merely been shifted a little. Is there a way to fix this? Physicists came up with a rather clever trick (I'm not sure whether it was mathematicians or physicists who first thought of it—in any case, I encountered it in a physics textbook): flip into a new dimension! The term they add is:
$$\int_{-1}^1 \frac{1}{x+i\varepsilon}dx$$
This moves the integrand out into the complex plane! The function is continuous everywhere along the real axis, so the Newton-Leibniz formula applies once again
$$\int_{-1}^1 \frac{1}{x+i\varepsilon}dx=\left.(\ln |x+i\varepsilon|)\right|_{-1}^1=0$$
The advantage of doing this is that, in the mathematical manipulation, the singularity is made to vanish, and the various infinities cancel out naturally, leaving behind the net result.
**This kind of manipulation is admittedly quite unrigorous, but in physics—especially in quantum field theory—when faced with all sorts of infinities, one struggles just to get a calculation done at all, let alone worry about rigor. Physicists always want to work out the answer first and check whether it matches reality, and only afterward worry about questions of rigor, axiomatization, and the like. So, as a product of this "transitional period," such computational tricks are absolutely necessary.
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