A) \[\frac{1}{5}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]
B) \[\frac{1}{9}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]
C) \[\frac{1}{4}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]
D) \[\frac{1}{9}\left[ \frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]
Correct Answer: B
Solution :
[b] Let \[I=\int\limits_{0}^{1}{\frac{dx}{({{x}^{2}}+16)({{x}^{2}}+25)}}\] \[=\frac{1}{9}\int\limits_{0}^{1}{\left( \frac{1}{{{x}^{2}}+16}-\frac{1}{{{x}^{2}}+25} \right)dx}\] \[=\frac{1}{9}\left( \frac{1}{4}{{\tan }^{-1}}\frac{x}{4}-\frac{1}{5}{{\tan }^{-1}}\frac{x}{5} \right)_{0}^{1}\] \[=\frac{1}{9}\left[ \frac{1}{4}{{\tan }^{-1}}\frac{1}{4}-\frac{1}{5}{{\tan }^{-1}}\frac{1}{5} \right]\]
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