A) \[\frac{x}{2}[sin(lo{{g}_{e}}x)-cos(lo{{g}_{e}}x)]+C\]
B) \[x[cos(lo{{g}_{e}}x)-\sin (lo{{g}_{e}}x)]+C\]
C) \[\frac{x}{2}[cos(lo{{g}_{e}}x)+\sin (lo{{g}_{e}}x)]+C\]
D) \[x[cos(lo{{g}_{e}}x)+\sin (lo{{g}_{e}}x)]+C\]
Correct Answer: C
Solution :
Let\[I=\int_{{}}^{{}}{\cos (lo{{g}_{e}}x)}dx\] \[\Rightarrow \]\[I=\cos (lo{{g}_{e}}x)x+\int_{{}}^{{}}{sin(lo{{g}_{e}}x)}dx\] \[\Rightarrow \]\[I=x\cos ({{\log }_{e}}x)+x\sin ({{\log }_{e}}x)x-\int_{{}}^{{}}{\cos ({{\log }_{e}}x)}dx\] \[\Rightarrow \]\[I=\frac{x}{2}[\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)]+C\]
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