I'm trying to do the following question: Spivak Chapter 14 Problem 25
The limit $\lim_{N\to\infty} \int_{a}^{\infty}f$, if it exists, is denoted by $\int_{a}^{\infty}f$, and called an "improper integral".
a) Determine $\int_{1}^{\infty}x^r$, if $r<-1$
c) Suppose that $f(x)\geq 0$ for all $x\geq 0$ and that $\int_{0}^{\infty}f$ exists. Prove that if $0\leq g(x)\leq f(x)$ for all $x\geq 0$, and g is integrable on each interval $[0,N]$, then $\int_{0}^{\infty}g$ also exists.
What I tried: For item a) I didn't have many ideas, the only that occurred to me was trying to partition the interval $[0, N]$ in equal intervals and try to fiddle with upper and lower sums, but as the exponent is negative I couldn't do much, and don't think that's the way. For item c) I believe that doing a) wouldn't help directly, so my idea was simply to use the fact that, as g is integrable on each interval $[0, N]$ the limit in the improper integral makes sense and thus $\int_{0}^{\infty}g$ would also exist, but that's most definitely wrong as I didn't rely on the other hypotheses, though I can't think of something else.
1] For your first question. You can first calculate $$ \int_{1}^{N}x^{r}\text{d}x=\left[\frac{x^{r+1}}{r+1}\right]^{N}_{1}=\frac{N^{r}}{r+1}-\frac{1}{r+1} $$ And $r<-1$ hence $$ \frac{N^{r}}{r+1} \underset{N \rightarrow +\infty}{\rightarrow}0 $$ So
In particular you have $$ \int_{1}^{+\infty}\frac{\text{d}x}{x^2}=1$$
2] The second is a little harder, consider the function $G$ given by $$ G\left(x\right)=\int_{0}^{x}g\left(t\right)\text{d}t $$ You know that $g \geq 0$ hence $G$ which is its antiderivative, is increasing. Furthermore for all $x$ $$ G\left(x\right) \leq \int_{0}^{+\infty}f\left(t\right)\text{d}t $$ It is increasing and is majorated, hence $G$ admits a limit. So $g$ is integrable. You can " adjust " it with sequences instead of functions which will fit more to the question.