A) \[\frac{1}{2}\log \frac{5}{3}\]
B) \[\frac{1}{3}\log \frac{5}{3}\]
C) \[\frac{1}{5}\log \frac{3}{5}\]
D) \[\frac{1}{2}\log \frac{3}{5}\]
Correct Answer: A
Solution :
Key Idea: If Integra; is in the form of\[\int{\frac{dx}{(ax+b)\sqrt{px+q}}}\], then put\[px+q={{t}^{2}}\], Let \[I=\int_{8}^{15}{\frac{dx}{(x-3)\sqrt{x+1}}}\] Put \[x+1={{t}^{2}}\Rightarrow dx=2t\,\,dt\] \[\therefore \] \[I=\int_{3}^{4}{\frac{2t\,\,dt}{({{t}^{2}}-4)t}}=2\int_{3}^{4}{\frac{dt}{{{t}^{2}}-4}}\] \[=2\times \frac{1}{2\times 2}\left[ \log \frac{t-2}{t+2} \right]_{3}^{4}\] \[=\frac{1}{2}\left( \log \frac{2}{6}-\log \frac{1}{5} \right)\] \[=\frac{1}{2}\log \frac{5}{3}\]
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