A) \[{{\text{e}}^{x}}cot\text{ }x+c\]
B) \[{{e}^{x}}\,log\text{ }sin\text{ }x+c\]
C) \[{{e}^{x}}log\text{ }sin\text{ }x+tan\text{ }x+c\]
D) \[{{e}^{x}}+sin\text{ }x+c\]
E) \[log(sin\,x+cos\,x)+{{e}^{x}}+c\]
Correct Answer: B
Solution :
We know\[\int{{{e}^{x}}[f(x)+f(x)]}dx={{e}^{x}}f(x)+c\] \[\therefore \]If\[f(x)=log\text{ }sin\text{ }x\] \[\Rightarrow \] \[f(x)=\frac{1}{\sin x}\cos x=\cot x\] \[\therefore \] \[\int{{{e}^{x}}(\log \sin x+\cot x)dx}={{e}^{x}}\log \sin x+c\]
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