A) \[{{e}^{x}}\log (\sec x)+c\]
B) \[{{e}^{x}}\log (\cos \text{ec}x)+c\]
C) \[{{e}^{x}}\log (\cos x)+c\]
D) \[{{e}^{x}}\log (\sin x)+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{{{e}^{x}}[\tan x-\log (\cos x)]\,dx}=\int_{{}}^{{}}{{{e}^{x}}[\tan x+\log (\sec x)]}\,dx\] \[={{e}^{x}}\log (\sec x)+c\] \[\left\{ \text{Since}\int_{{}}^{{}}{{{e}^{x}}\left\{ f(x)+{f}'(x) \right\}dx={{e}^{x}}f(x)+c} \right\}\].
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