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# Improper Integrals questions - converge or diverge?

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

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

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

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

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

I don't know how to integrate this

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• 1 month later...

For (1) I got :

$\int \frac{1}{x\ln(x)}dx\underbrace{=}_{y=\ln(x),dy=dx/x}\int\frac{dy}{y}=\ln(y)=\ln(\ln(x))$

However, I don't know if it gives the same as your expression.

(2) doesn't it give sthg. proto. :

$\ln\left(\frac{x-2}{x+2}\right)$ ?

(3) I derived :

$\left(\frac{x}{\ln(x)}\right)'=\frac{\ln(x)-1}{\ln(x)^2}$

which looks near to the result, but it seems not to be the same.

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