Hi all,
Are all functions 'Integrable'? Is there any function existing which is not integrable? can anyone give me an example??
No, a function is not defined if it has an infinite amount of discontiunities on the interval it is being integrated on..
Also I could interpret this as
Neither
$\displaystyle \int_0^{1}\frac{dx}{x^2}$ or $\displaystyle \int_1^{\infty}\frac{dx}{\sqrt{x}}$
are "not integrable" because they diverge...
it all depends on how you define integrable
No. Dirichlet's function is an example.
That is wrong. A function can have infinitely many discontinouties and still be integrable.
Your examples with improper integrals are bad examples because "integrable" is defined on closed bounded intervals.it all depends on how you define integrable
This is the case with Laplace Transforms, right (the functions must be piecewise continuous in order to apply the transform)?
Also, I believe $\displaystyle \int e^xtan(x)\,dx$ is another integral that can't be solved...
That integral came up in my Differential Equations class when solving $\displaystyle y^{//}+y=tan(x)$ (this was the correct equation). He gave us $\displaystyle y^{//}-y=tan(x)$, and we wasted 45 minutes trying to figure that one out... :'(
I do not know the theory behind Laplace transforms so I cannot say. But I am assuming that the necessacity of "piecewise continous" can be relaxed and you can use "integrable" instead. The reason why books do not use that is because they are engineering books and engineering students do not know what "integrable" means. Also, Laplace Transforms are not a good example for the above question because the poster was asking about proper integrables, i.e. integrals over finite closed intervals.
I think Chris meant that you cannot find the integral in closed form. But "integrable" does not mean having a closed form. In fact, "integrable" is used only for definite integrals. Integrable means that the Riemann Sums approximate the function well. So we can take a limiting value and call that to be the integral. Though not always it is possible.