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Evaluate the improper integral:

$$\int_2^\infty \frac{e^{-2x}}{6-e^{-2x}}dx$$

A

$\frac{1}{2}\left(\ln 6 - \ln(6-e^{-4})\right)$

B

$\ln 3 - \ln(3-\frac{1}{2}e^{-4})$

C

$\frac{1}{6}\left(\ln 6 - \ln(6-e^{-4})\right)$

D

$\frac{1}{3}\left(\ln 2 - \ln(2-e^{-2})\right)$

E

infinite integral

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