For an improper integral of the form $$\int_{-\infty}^{\infty}f(x) \, dx,$$
I'm told that I must set $$\int_{-\infty}^{\infty}f(x) \, dx =\lim_{c \to \infty} \int_k^cf(x) \, dx+\lim_{c \to -\infty}\int_c^kf(x) \, dx.$$
Why can I not set $$\int_{-\infty}^{\infty}f(x) \, dx =\lim_{a\to \infty}\int_{-a}^a f(x) \, dx \, ?$$
The reason for the definition is:
It could happen that $$ \lim_{a\to \infty}\int_{-a}^a f(x) \, dx \tag1$$ exists but one or both of $$ \lim_{c \to \infty} \int_k^cf(x)\; dx,\qquad\lim_{c \to -\infty}\int_c^kf(x) \, dx. \tag2$$ do not exist.
The situation where both of $(2)$ exist is called "convergence" of the improper integral. The situation where $(1)$ exists is called the "principal value" of the integral. The limit in $(1)$ may sometimes be written $$ \text{P.V. }\int_{-\infty}^\infty f(x)\;dx . $$
The principal value can have bad properties that a convergent integral cannot.
examples
$$ \text{P.V. }\int_{-\infty}^\infty x \; dx = 0 $$ change variables $y=x+1$ to get $$ \text{P.V. }\int_{-\infty}^\infty (y+1) \; dy = \infty $$