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J & K CET Engineering J and K - CET Engineering Solved Paper-2005

  • question_answer
    The value of \[\int_{1}^{{{e}^{2}}}{\frac{dx}{x{{(1+\log \,x)}^{2}}}}\] is

    A)  \[2/3\]           

    B)  \[1/3\]

    C)  \[3/2\]           

    D)  In 2               

    Correct Answer: A

    Solution :

    Let \[I=\int_{1}^{{{e}^{2}}}{\frac{dx}{x{{(1+\log x)}^{2}}}}\] Put \[1+\log \,x=t\] \[\Rightarrow \] \[\frac{1}{x}dx=dt\] when \[x=1,\,t=1\] and \[x={{e}^{2}},t=3\] \[\therefore \] \[I=\int_{1}^{3}{\frac{1}{{{t}^{2}}}dt}\] \[=\left[ -\frac{1}{t} \right]_{1}^{3}=-\frac{1}{3}+1=\frac{2}{3}\]


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