$\int_5^{10} \frac{d x}{(x-1)(x-2)}=$

$\int_5^{10} \frac{d x}{(x-1)(x-2)}=$
  1. $\log \left|\frac{27}{32}\right|$
  2. $\log \left|\frac{3}{4}\right|$
  3. $\log \left|\frac{8}{9}\right|$
  4. $\log \left|\frac{32}{27}\right|$

Solution

$\begin{aligned} & I=\int_5^{10} \frac{d x}{(x-1)(x-2)} \\ & =\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right](-1) d x=-\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right] d x \\ & =-[\log |x-1|]_5^{10}+[\log |x-2|]_5^{10}=-[\log |9|-\log |4|]+[\log |8|-\log |3|] \\ & =\left[\log \left|\frac{8}{3}\right|\right]-\left[\log \left|\frac{9}{4}\right|\right]=\log \left|\frac{8}{3} \times \frac{9}{4}\right|=\log \left|\frac{32}{27}\right|\end{aligned}$

Asked in: MHT CET 2021 (21 Sep Shift 2)

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