A) \[{{e}^{x}}\cot x\]
B) \[{{e}^{x}}\tan x\]
C) \[{{e}^{x}}\sec x\]
D) \[{{e}^{x}}\cos x\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{{{e}^{x}}(1+\tan x)\sec x\,dx}=\int_{{}}^{{}}{{{e}^{x}}\sec x\,dx}+\int_{{}}^{{}}{{{e}^{x}}\tan x\sec x\,dx}\] \[={{e}^{x}}\sec x-\int_{{}}^{{}}{{{e}^{x}}\sec x\tan x\,dx}+\int_{{}}^{{}}{{{e}^{x}}\sec x\tan x\,dx}\] \[={{e}^{x}}\sec x+c.\] Aliter : \[\int_{{}}^{{}}{{{e}^{x}}(\sec x+\sec x\tan x)\,dx}={{e}^{x}}\sec x+c\] Obviously, it is of the form \[\int_{{}}^{{}}{{{e}^{x}}\left\{ f(x)+{f}'(x) \right\}}\,dx.\]
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