A) \[\log (x-3)-\log (x-2)+c\]
B) \[\log {{(x-3)}^{2}}-\log (x-2)+c\]
C) \[\log (x-3)+\log (x-2)+c\]
D) \[\log {{(x-3)}^{2}}+\log (x-2)+c\]
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\frac{x-1}{(x-3)(x-2)}\,dx}\] \[=\int_{{}}^{{}}{\frac{x-3}{(x-3)(x-2)}\,dx+\int_{{}}^{{}}{\frac{2}{(x-3)(x-2)}}}\,dx\] \[=\log \left[ \frac{(x-2){{(x-3)}^{2}}}{{{(x-2)}^{2}}} \right]+c=\log \left[ \frac{{{(x-3)}^{2}}}{(x-2)} \right]+c.\] Trick : By inspection, \[\frac{d}{dx}\left\{ \log (x-3)-\log (x-2) \right\}\] \[=\frac{1}{x-3}-\frac{1}{x-2}=\frac{1}{(x-3)(x-2)}\] \[\Rightarrow \frac{d}{dx}\left\{ 2\log (x-3)-\log (x-2) \right\}\] \[=\frac{2}{x-3}-\frac{1}{x-2}=\frac{x-1}{(x-3)(x-2)}\].
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