Improper Integrals - converge or diverge

[zeion]

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) $$\int_{e}^{\infty} \frac{dx}{xlnx}$$



2) $$\int_{1}^{4} \frac{dx}{x^2 - 4}$$



3) $$\int_{e}^{\infty} \frac{dx}{(lnx)^2}$$



4) $$\int_{2}^{\infty} \frac {dx}{x^2 + sinx}$$

Homework Equations

The Attempt at a Solution

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

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

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

I sub in and get ln0?

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

4) I don't know how to integrate this

 

[LCKurtz]

The Attempt at a Solution

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

Your antiderivative is incorrect. Try u = ln(x)

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

Don't you need a + sign between them?

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

Where did $c \rightarrow \infty$ come from? Your intgerals are improper at x = 2.

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

No.

4) I don't know how to integrate this

To integrate 1/(x-a) try a u substitution.