Originally Posted by

**Jacobpm64** Use the box and the behavior of rational and exponential functions as $\displaystyle x \rightarrow \infty $ to predict whether the integrals converge or diverge.

Here is the box:

$\displaystyle \int^\infty_1 \frac{1}{x^p} dx $ converges for p > 1 and diverges for p __<__ 1.

$\displaystyle \int^1_0 \frac{1}{x^p} dx $ converges for p < 1 and diverges for p __>__ 1.

$\displaystyle \int^\infty_0 e^{-ax} dx $ converges for a > 0.

Here is the problem I need help with .. along with my work:

$\displaystyle \int^\infty_1 \frac{x^2+1}{x^3 + 3x + 2} dx $

I know that this integral is less than $\displaystyle \int^\infty_1 \frac{1}{x} dx $. I also know that $\displaystyle \int^\infty_1 \frac{1}{x} dx $ diverges. This does not help me though because I can not use a diverging integral to say that a smaller integral is also diverging. This is where I'm confused.