TITLE: What is the integral of a function from $\infty$ to $\infty$?
QUESTION [6 upvotes]: I apologize in advance for my bad English, I am italian.
I took a Calculus exam today, and one of the exercises was:
$$ \lim_{x\to\infty} \int_x^{x+1/x}e^{t^2}dt $$
I answered $0$ even though I knew that $\int_x^xf(x)dx=0$ is only true for a finite 
$x$, in fact I was wrong.
I could not ask the teacher for the right answer and I still can't see a way to calculate this limit.
PS: I guess my exam was pretty similar to the one you call Calculus I, so please do not use advanced theorems.

REPLY [18 votes]: $f(t)=e^{t^2}$ is an increasing function over $\mathbb{R}^+$, hence
$$\int_{x}^{x+\frac{1}{x}} e^{t^2}\,dt \geq \int_{x}^{x+\frac{1}{x}}e^{x^2}\,dt = \frac{e^{x^2}}{x} $$
and the limit is just $+\infty$.