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## Homework Statement

Use the box and the behavior of rational and exponential functions as [tex] x \rightarrow \infty [/tex] to predict whether the integrals converge or diverge.

Here is the box:

[tex] \int^\infty_1 \frac{1}{x^p} dx [/tex] converges for p > 1 and diverges for p

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[tex] \int^1_0 \frac{1}{x^p} dx [/tex] converges for p < 1 and diverges for p

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[tex] \int^\infty_0 e^{-ax} dx [/tex] converges for a > 0.

Problem 1:

[tex] \int^\infty_1 \frac{x^2}{x^4 + 1} dx [/tex]

Problem 2:

[tex] \int^\infty_1 \frac{x^2 - 6x + 1}{x^2 + 4} dx [/tex]

## Homework Equations

The ones in the box above.

## The Attempt at a Solution

Problem 1:

I know that this integral is less than [tex] \int^\infty_1 \frac{1}{x} dx [/tex]. I also know that [tex] \int^\infty_1 \frac{1}{x} dx [/tex] 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.

Problem 2:

I know that this integral is less than [tex] \int^\infty_1 1 dx [/tex] I also know that [tex] \int^\infty_1 1 dx [/tex] diverges. This does not help me though because I can not use a diverging integral to say that a smaller integral is also diverging. So, I'm confused at the same place on this problem.