Improper integrals Question

Question

I was asked to define the next intergrals and I want to know if I did it right:
$$1) \int^\infty_a f(x)dx = \lim_{b \to \infty}\int^b_af(x)dx$$ $$2) \int^b_{-\infty} f(x)dx = \lim_{a \to -\infty}\int^b_af(x)dx$$ $$3) \int^\infty_{-\infty} f(x)dx = \lim_{b \to \infty}\int^b_0f(x)dx + \lim_{a \to -\infty}\int^0_af(x)dx$$ Thanks.

Answer 1

The first two definitions you gave are the standard definitions, for $f$ say continuous everywhere. The third is more problematical, It is quite possible that the definition in your course is $$\lim_{a\to-\infty, b\to \infty} \int_a^b f(x)\,dx.$$ So $a\to-\infty$, $b\to\infty$ independently.

What you wrote down would then be a fact rather than a definition.

Answer 2

In some cases, such as Fourier transformation, it is critical that we have

$$\int_{-\infty}^{\infty} dx \, f(x) = \lim_{a \to \infty} \int_{-a}^{a} dx \, f(x)$$