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## Homework Statement

Hello. I have some questions on this assignment, I'm wondering if I could get some help:

Determine whether the integral converges and, if so, evaluate the integral.

1) [tex] \int_{e}^{\infty} \frac{dx}{xlnx} [/tex]

2) [tex] \int_{1}^{4} \frac{dx}{x^2 - 4} [/tex]

3) [tex]\int_{e}^{\infty} \frac{dx}{(lnx)^2}[/tex]

4) [tex]\int_{2}^{\infty} \frac {dx}{x^2 + sinx} [/tex]

## Homework Equations

## The Attempt at a Solution

1) I integrate and get [tex] \lim_{b \to \infty} \int_{e}^{b} \frac{dx}{xlnx} = \lim_{ b \to \infty} [ \frac{ln(xlnx)}{lnx}]_{e}^{b} [/tex] ?

2) It has discontinuity at x = 2 and x = -2 so I evaluate [tex] \int_{1}^{2} \frac{dx}{x^2 - 4} \int_{2}^{4} \frac{dx}{x^2 - 4} [/tex]

I integrate and get [tex] \frac{-1}{4} \lim_{c \to \infty} \int_{1}^{c} \frac{1}{x+2} - \frac{1}{x-2} dx [/tex]

I sub in and get ln0?

3) Does this integrate into [tex] \lim_{b \to \infty} [\frac {-x}{lnx}]_{e}^{b} [/tex] ?

4) I don't know how to integrate this