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KVPY Sample Paper KVPY Stream-SX Model Paper-23

  • question_answer
    If \[{{I}_{n}}=\int_{1}^{e}{{{(\ln x)}^{n}}dx,n\in N,}\] then \[{{I}_{10}}+10\,{{I}_{9}}\] is equal to

    A) \[{{e}^{10}}\]

    B) \[\frac{{{e}^{10}}}{10}\]

    C) \[e\]

    D)  \[e-1\]

    Correct Answer: C

    Solution :

    We have,
    \[{{I}_{n}}=\int\limits_{1}^{e}{{{(\ln x)}^{n}}dx}\]\[\Rightarrow \]\[{{I}_{n}}=[{{(\ln x)}^{n}}x]_{1}^{e}-\int\limits_{1}^{e}{\frac{n\,{{(\ln x)}^{n-1}}}{x}}\cdot x\,dx\]\[\Rightarrow \]\[{{I}_{n}}=e-n{{I}_{n-1}}\]\[\Rightarrow \]\[{{I}_{n}}+n{{I}_{n-1}}=e\]
    \[\therefore \]\[{{I}_{10}}+10{{I}_{9}}=e\]


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