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JEE Main & Advanced Mathematics Definite Integration Question Bank Fundamental definite integration, Definite integration by substitution

  • question_answer
    If \[{{I}_{1}}=\int_{e}^{{{e}^{2}}}{\frac{dx}{\log x}}\] and \[{{I}_{2}}=\int_{1}^{2}{\frac{{{e}^{x}}}{x}\,dx,}\] then                                                                 [Karnataka CET 2000]

    A)                 \[{{I}_{1}}={{I}_{2}}\]    

    B)                 \[{{I}_{1}}>{{I}_{2}}\]

    C)                 \[{{I}_{1}}<{{I}_{2}}\]    

    D)                 None of these

    Correct Answer: A

    Solution :

               Put \[\log x=u\]in \[{{I}_{1}},\]so that \[dx=x\,du={{e}^{u}}du\]            Also as \[x=e\]to \[{{e}^{2}},u=1\]to 2                                 Thus, \[{{I}_{1}}=\int_{1}^{2}{\frac{{{e}^{u}}}{u}du=\int_{1}^{2}{\frac{{{e}^{x}}}{x}dx}}\]. Hence, \[{{I}_{1}}={{I}_{2}}\].


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