A) \[x\,\,{{e}^{\sin x}}-{{e}^{\sin x}}\sec x+C\]
B) \[x\,\,{{e}^{\cos x}}-{{e}^{\sin x}}\sec x+C\]
C) \[{{x}^{2}}{{e}^{\sin x}}+{{e}^{\sin x}}\sec x+C\]
D) \[2x\,\,{{e}^{\sin x}}-{{e}^{\sin x}}\tan x+C\]
Correct Answer: A
Solution :
\[\int{{{e}^{\sin x}}}\left( \frac{x{{\cos }^{3}}x-\sin x}{{{\cos }^{2}}x} \right)dx\] \[=\int{{{e}^{\sin x}}}x\cos x\,\,dx-\int{{{e}^{\sin x}}}\tan x\sec x\,\,dx\] \[=\int{x\,\,d}\left( {{e}^{\sin x}} \right)-\int{{{e}^{\sin x}}}d(\sec x)\] \[=\left\{ x\,\,e\,\,{{\sin }^{x}}-\int{{{e}^{\sin x}}dx} \right\}\] \[-\left\{ {{e}^{\sin x}}\sec x-\int{{{e}^{\sin x}}\sec x\cos x\,\,dx} \right\}\] \[=x\,\,{{e}^{\sin x}}-{{e}^{\sin x}}\sec x+C\]
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