A) \[({{e}^{x}}.\cos ecx)+C\]
B) \[{{e}^{x}}\cot x+C\]
C) \[({{e}^{x}}.\sec x)+C\]
D) \[{{e}^{x}}\tan x+C\]
Correct Answer: C
Solution :
\[I=\int{{{e}^{x}}\left( \frac{\sin x+\cos x}{1-{{\sin }^{2}}x} \right)dx}\] \[I=\int{{{e}^{x}}\left( \frac{\sin x+\cos x}{{{\cos }^{2}}x} \right)dx}\] \[I=\int{{{e}^{x}}\,\tan x.\sec x+dx+\int{\underset{II}{\mathop{{{e}^{x}}}}\,.\underset{I}{\mathop{\sec x}}\,\,\,dx}}\] \[I=\int{{{e}^{x}}\tan x.\sec x\,dx+(\sec \,x\,.\,{{e}^{x}}}\] \[-\int{\sec x.\tan x\,{{e}^{x}}}dx\}\] \[I=\int{{{e}^{x}}\,\tan x.\sec x\,\,dx}+{{e}^{x}}\sec x\] \[-\int{{{e}^{x}}\,\tan \,x.\,\sec x\,dx}\] \[I={{e}^{x}}.\,\sec \,\,x+c\]
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