Question

$\int\limits_{5}^{10} \frac{1}{\left(x-1\right)\left(x-2\right)}dx$ is equal to

• A. $log \frac{27}{32}$
• B. $log \frac{32}{27}$
• C. $log \frac{8}{9}$
• D. $log \frac{3}{4}$

Answer 1

Answer: (B)

$log \frac{32}{27}$

Answer 2

Let $I =\int \limits_{5}^{10} \frac{1}{(x-1)(x-2)} d x$ $=\int \limits_{5}^{10}\left[\frac{-1}{x-1}+\frac{1}{x-2}\right] d x$ $=[-\log (x-1)+\log (x-2)]_{5}^{10}$ $=-\log 9+\log 8+\log 4-\log 3$ $=-2 \log 3+3 \log 2+2 \log 2-\log 3$ $=-3 \log 3+5 \log 2$ $=-\log 27+\log 32$ $=\log \frac{32}{27}$