# Thread: Are these functions integrable?

1. ## Are these functions integrable?

I've recently been shown some of the standard ways used in order to deduce whether or not a function is integrable over a given interval (eg. comparison, monotone convergence theorem, bounded functions on bounded intervals etc), but I'm not sure how to actually spot these kind of things in actual questions. For example, what can we use to show that the following functions are integrable:

1) $\displaystyle x^n \exp(-x)$ over $\displaystyle (0,\infty)$

2) $\displaystyle x^a \log x$ over (0,1] for a > -1

And in case 2 above, how would we deduce that
$\displaystyle x^b (1-x)^{-1} \log x$ is intergable over (0,1] for b > -1?

2. 1) $\displaystyle x^n \exp(-x)$ over $\displaystyle (0,\infty)$

We know that $\displaystyle \lim_{x \to +\infty}\:x^{n+2}\:e^{-x} = 0$

Therefore $\displaystyle \exists A>0 \:\:\forall x>A\:\: |x^{n}\:e^{-x}| \leq \frac{1}{x^2}$

And $\displaystyle \frac{1}{x^2}$ is integrable over $\displaystyle (A,+\infty)$