A) \[\log \left( \frac{8}{9} \right)\]
B) \[\log \left( \frac{9}{8} \right)\]
C) \[\log (8\times 9)\]
D) None of these
Correct Answer: B
Solution :
Let \[I=\int_{0}^{\pi /2}{\frac{\sin x\cos x.dx}{{{\cos }^{2}}x+3\cos x+2}}\] We put \[\cos x=t\Rightarrow -\sin x\,dx=dt,\]then \[I=\int_{0}^{1}{\frac{t.dt}{{{t}^{2}}+3t+2}=\int_{0}^{1}{\left[ \frac{2}{t+2}-\frac{1}{t+1} \right]}}\,dt\] \[=[2\log (t+2)-\log (\,t+1)]_{0}^{1}\]\[=[2\log 3-\log 2-2\log 2]\] \[=[2\log 3-3\log 2]=[\log 9-\log 8]=\log \left( \frac{9}{8} \right)\].
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