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Improper Integrals - converge or diverge

  • Thread starter zeion
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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

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
LCKurtz
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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] ?
Your antiderivative is incorrect. Try u = ln(x)

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]
Don't you need a + sign between them?

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]
Where did [itex]c \rightarrow \infty[/itex] come from? Your intgerals are improper at x = 2.
3) Does this integrate into [tex] \lim_{b \to \infty} [\frac {-x}{lnx}]_{e}^{b} [/tex] ?
No.

4) I don't know how to integrate this
To integrate 1/(x-a) try a u substitution.
 

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