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J & K CET Engineering J and K - CET Engineering Solved Paper-2012

  • question_answer
    If \[f(x)={{\log }_{e}}\,(1+x)-{{\log }_{e}}(1-x),\]then the value of \[\int_{-1/2}^{1/2}{f(x)\,\,\,dx}\] equals to

    A)  \[0\]                 

    B)  \[1\]

    C)  \[\frac{1}{2}\]                

    D)  \[-\frac{1}{2}\]

    Correct Answer: A

    Solution :

    Given, \[f(x)={{\log }_{e}}(1+x)-{{\log }_{e}}(1-x)\] \[\therefore \] \[f(-x)={{\log }_{e}}(1-x)-{{\log }_{e}}(1+x)\] \[=-[{{\log }_{e}}(1+x)-{{\log }_{e}}(1-x)]\] \[=-f(x)\] \[\therefore \] \[f(x)\] is an odd function. \[\therefore \] \[\int_{-1/2}^{1/2}{f(x)\,dx=0}\]


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