Hi, I'm having some difficulty in recognising what new improper integral to create when doing a comparison test.

IS there a step-by-step logical approach that could be used?

My understanding is, we do comparison tests because the improper integral being dealt with is too difficult to integrate. I understand the rules in that if f(x)>g(x)

then f(x) is divergent if g(x) is divergent and g(x) is convergent if f(x) is convergent.

But, I don't understand a few problems:

So, for example,

http://i.imgur.com/OF9Ip.jpg

If you could look at 6b, and the solutions to 6b.

I don't understand what the solutions mean when they say "Use x< x^1/4"

The same with 6c. Why have they said 'use 1/3y^4' ?

I managed to work out 6a myself. are there flaws in the solutions? What am I missing here..

Additionally, in regards to the matrices question at the top, what method is being referred to there? Is it asking for me to use row operations?

Thanks!