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JEE Main & Advanced Sample Paper JEE Main Sample Paper-7

  • question_answer
    If \[\int\limits_{{}}^{{}}{\frac{\cos x-\sin x+1-x}{{{e}^{x}}+\sin x+x}}dx=\ell n\,\left( fx \right)+g(x)+C,\]  where C is the constant of integration and \[f(x)\] is positive, then \[f(x)+g(x)\] has the value equal to

    A)  \[{{e}^{x}}+\sin x+2x\]

    B)  \[{{e}^{x}}+\sin x\]

    C)  \[{{e}^{x}}-\sin x\]                        

    D)  \[{{e}^{x}}+\sin x+x\]

    Correct Answer: B

    Solution :

    \[I\int_{{}}^{{}}{\frac{\left( {{e}^{x}}+\cos x+1 \right)-\left( {{e}^{x}}+\sin x+x \right)}{{{e}^{x}}+\sin x+x}dx}\] \[\Rightarrow \,\,I=In\left( {{e}^{x}}+\sin x+x \right)-x+c\] \[\therefore \,f(x)={{e}^{x}}+\sin x+x\] and \[g(x)=-x\] \[\therefore \,\,f(x)+g(x)={{e}^{x}}+\sin x\].


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