A) \[{{\log }_{e}}\frac{{{(x-2)}^{2}}}{(x-1)}+p\]
B) \[{{\log }_{e}}\frac{(x-1)}{(x-2)}+p\]
C) \[\frac{x-1}{x-2}+p\]
D) \[2{{\log }_{e}}\left( \frac{x-2}{x-1} \right)+p\] (where p is an arbitrary constant)
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{x}{(x-2)(x-1)}\,dx}=-\int_{{}}^{{}}{\frac{1}{x-1}\,dx+\int_{{}}^{{}}{\frac{2}{x-2}\,dx}}\] \[=-{{\log }_{e}}(x-1)+2{{\log }_{e}}(x-2)+c={{\log }_{e}}\frac{{{(x-2)}^{2}}}{(x-1)}+p.\]
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