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JEE Main & Advanced Sample Paper JEE Main Sample Paper-22

  • question_answer
    The value of \[\underset{a\to \infty }{\mathop{\lim }}\,\frac{1}{{{a}^{2}}}\int\limits_{0}^{a}{ln(1+{{e}^{x}})dx}\]  equals

    A)  0                                

    B)  1

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

    D)  non-existent

    Correct Answer: C

    Solution :

    \[\underset{x\to \infty }{\mathop{Lim}}\,\frac{1}{{{a}^{2}}}\,\int\limits_{0}^{a}{\ln \,(1+{{e}^{x}})\,dx,}\] we get             Using Leibnitz Rule \[\underset{a\to \infty }{\mathop{Lim}}\,\,\frac{\ln (1+{{e}^{a}})}{2a}\,\left( \frac{\infty }{\infty } \right)=\frac{1}{2}\,\underset{x\to \infty }{\mathop{Lim}}\,\,\frac{{{e}^{a}}}{1+{{e}^{a}}}\,=\frac{1}{2}\]


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