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Non-Integrable Functions

How is a function ascertained to be integrable or non integrable (indefinite integration). For example, if you are provided a function $$f(x)$$, what steps would you take to check if it is integrable or not.

Note by Bruce Wayne
3 years, 9 months ago

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It depends whether you mean "does the function have an integral?" or "does the function have an integral that I can express in terms of standard functions?". The first question is much easier to answer. There are (fundamentally) two theories of integration, Riemann and Lebesgue Integration, and each have their definitions for what it means to be integrable, and these conditions can be checked for a particular function. For both integrals, for example, it is easy to show that any continuous function is integrable.

Whether an indefinite integral can be expressed in terms of "standard" functions is much harder, and it all depends on what you mean by "standard". We cannot express the indefinite integral $\tfrac{1}{\sqrt{2\pi}}\int e^{-\frac12x^2}\,dx$ in terms of "elementary" functions (powers, trig functions, logarithms, exponentials). One solution to this problem is to define a new "standard" function to do the job, namely $\Phi(x) \; =\; \tfrac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac12u^2}\,du$ In mathematics, you can always define a new function if you need it! The question is whether your new function is sufficiently interesting to be accepted as "standard". Given the central important of the Normal distribution in statistics, the function $$\Phi$$ is well accepted, for example. · 3 years, 9 months ago