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JEE Main & Advanced Sample Paper JEE Main Sample Paper-15

  • question_answer
    Let\[{{I}_{n}}=\int\limits_{0}^{1}{{{(x\ln x)}^{n}}dx,}\] if\[{{I}_{4}}=k\int\limits_{0}^{1}{{{x}^{4}}\ln {{x}^{3}}dx,}\]then ?k? is equal to

    A) \[\frac{-2}{3}\]                                 

    B) \[\frac{-4}{5}\]

    C) \[\frac{2}{3}\]                                   

    D) \[\frac{-5}{6}\]

    Correct Answer: B

    Solution :

    \[{{I}_{n}}=\int\limits_{0}^{1}{{{(x\,\ln \,x)}^{n}}dx}\] \[{{I}_{n}}=\left. \left( {{\left( \ln x \right)}^{n}}.\frac{{{x}^{n+1}}}{n+1} \right) \right|_{0}^{1}-\frac{n}{n+1}\int\limits_{0}^{1}{{{x}^{n}}{{(\ln x)}^{n-1}}dx}\] \[\therefore \]\[{{I}_{4}}=\frac{-4}{5}\int\limits_{0}^{1}{{{x}^{4}}\ln {{x}^{3}}dx}\]


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