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I wanna prove that the following improper integral converge but I dont't have any clear idea of how to do this:

$$\int_3^\infty \frac{4}{4x \ln^2 (y)} \ dx$$

From what I have learned, usually improper integral could be separated it into two parts, but I don't whether I can apply that method for the improper integral. Does someone have any thoughts?

The Pointer
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Birk Lundgren
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2 Answers2

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hint

Your integral has the same nature than

$$J=\int_e^\infty\frac{dx}{x\ln^2(x)}$$

now, make the substitution $$t=\ln(x)$$ with

$$dt=\frac{dx}{x}$$

So, $ J $ has the same convergence than

$$\int_1^\infty \frac{dt}{t^2}$$ which clearly converges.

hamam_Abdallah
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This integral is improper because of the infinite upper limit, not because of the lower limit. The integrand is defined and finite for all $x \geq 2$. Therefore, you need only replace the upper limit of integration with a finite limit (call it $b$), evaluate the resulting proper integral, then take the limit of the answer as $b$ approaches infinity: $$\displaystyle\int_2^\infty \frac{1}{x \ln^2 x} \: dx = \lim_{b \to \infty} \int_2^b \frac{1}{x \ln^2 x} \: dx$$

KenM
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