MHT CET · Maths · Definite Integration
\(\int_5^{10} \frac{d x}{(x-1)(x-2)}=\)
- A \(\log \left|\frac{27}{32}\right|\)
- B \(\log \left|\frac{3}{4}\right|\)
- C \(\log \left|\frac{8}{9}\right|\)
- D \(\log \left|\frac{32}{27}\right|\)
Answer & Solution
Correct Answer
(D) \(\log \left|\frac{32}{27}\right|\)
Step-by-step Solution
Detailed explanation
\(I=\int_5^{10} \frac{d x}{(x-1)(x-2)} \)
\( =\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right](-1) d x=-\int_5^{10}[\frac{1}{x-1}-\) \(\frac{1}{x-2}] d x \)
\( =-[\log |x-1|]_5^{10}+[\log |x-2|]_5^{10}=-\) \([\log |9|-\log |4|]~+\) \([\log |8|-\log |3|] \)
\( =\left[\log \left|\frac{8}{3}\right|\right]-\left[\log \left|\frac{9}{4}\right|\right]=\log \left|\frac{8}{3} \times \frac{9}{4}\right|=\log \left|\frac{32}{27}\right|\)
\( =\int_5^{10}\left[\frac{1}{x-1}-\frac{1}{x-2}\right](-1) d x=-\int_5^{10}[\frac{1}{x-1}-\) \(\frac{1}{x-2}] d x \)
\( =-[\log |x-1|]_5^{10}+[\log |x-2|]_5^{10}=-\) \([\log |9|-\log |4|]~+\) \([\log |8|-\log |3|] \)
\( =\left[\log \left|\frac{8}{3}\right|\right]-\left[\log \left|\frac{9}{4}\right|\right]=\log \left|\frac{8}{3} \times \frac{9}{4}\right|=\log \left|\frac{32}{27}\right|\)
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