JEE Main & Advanced Sample Paper JEE Main Sample Paper-20

  • question_answer
    Find\[\int{{{e}^{\sin x}}}\left( \frac{x{{\cos }^{3}}x-\sin x}{{{\cos }^{2}}x} \right)dx\]

    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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