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Question: Answered & Verified by Expert
$\int_5^{10} \frac{d x}{(x-1)(x-2)}=$
MathematicsDefinite IntegrationMHT CETMHT CET 2021 (21 Sep Shift 2)
Options:
  • A $\log \left|\frac{27}{32}\right|$
  • B $\log \left|\frac{3}{4}\right|$
  • C $\log \left|\frac{8}{9}\right|$
  • D $\log \left|\frac{32}{27}\right|$
Solution:
1752 Upvotes Verified Answer
The correct answer is: $\log \left|\frac{32}{27}\right|$
$\begin{aligned} & I=\int_5^{10} \frac{d x}{(x-1)(x-2)} \\ & =\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right](-1) d x=-\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right] d x \\ & =-[\log |x-1|]_5^{10}+[\log |x-2|]_5^{10}=-[\log |9|-\log |4|]+[\log |8|-\log |3|] \\ & =\left[\log \left|\frac{8}{3}\right|\right]-\left[\log \left|\frac{9}{4}\right|\right]=\log \left|\frac{8}{3} \times \frac{9}{4}\right|=\log \left|\frac{32}{27}\right|\end{aligned}$

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