Integrating odd functions with infinite discontinuity:

[atqamar]

If an odd function has an infinite discontinuity in its domain, can it be integrated (such that a convergent finite emerges) with that domain included?

For example: $$\int_{-1}^2 \frac{1}{x^{-3}} dx$$. Intuitively, it can be simplified to $$\int_1^2 \frac{1}{x^{-3}} dx$$ and thus the infinite discontinuity at 0 is removed.

If that is not doable, can an integral converge if the end points of the domain are infinite discontinuities?

For example: Does $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} tan(x) dx = 0$$?

If these kinds of functions are split in two, and limits to $$\infty$$ are taken, then algebraic manipulation of infinities are required.

Any insight would be appreciated.

 

[NoMoreExams]

You should look into improper integrals?

 

[atqamar]

Certainly, these are all improper integrals. But my question is whether the above integrals would converge or diverge. Intuitively/Geometrically, they should converge; but like I said, once the improper integral is carried out, one has to deal with infinities.

 

[Gib Z]

Look up "Cauchy Principal Value". Strictly, the integrations over the discontinuities are divergent, but they can be assigned a Cauchy Principal Value that appeals to your geometric intuition.

 

[atqamar]

Thanks a lot Gib Z! That helped immensely.