A) 1
B) \[\frac{1}{2}\]
C) 0
D) \[\frac{1}{101}\]
Correct Answer: C
Solution :
Let\[I=\int_{1/2}^{2}{\frac{1}{x}\cos e{{c}^{101}}\left( x-\frac{1}{x} \right)}dx\] put \[x=\frac{1}{t}\] \[\Rightarrow \] \[I=\int_{2}^{1/2}{t.\cos e{{c}^{101}}\left( \frac{1}{t}-t \right)\left( \frac{1}{{{t}^{2}}} \right)}dt\] \[=-\int_{2}^{1/2}{\frac{1}{t}\cos e{{c}^{101}}\left( t-\frac{1}{t} \right)}dt\] \[\Rightarrow \] \[I=-I\] \[\Rightarrow \] \[2I=0\] \[\therefore \] \[I=0\]
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