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  3. 12th Class
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  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If \[{{I}_{m}}=\int_{1}^{x}{{{(\log x)}^{m}}dx}\] satisfies the relation \[{{I}_{m}}=k-l{{I}_{m-1}},\] then

    A) \[k=e\]                                   

    B) \[l=m\]

    C) \[k=\frac{1}{e}\]                

    D) None of these

    Correct Answer: B

    Solution :

    • \[{{I}_{m}}=\int_{1}^{x}{{{(\log x)}^{m}}dx}=({{(\log x)}^{m}}.x)_{1}^{x}-\int_{1}^{x}{m{{(\log x)}^{m-1}}.\frac{1}{x}x\,\,dx}\]               
    • \[={{(\log x)}^{m}}.x-m{{I}_{m-1}}\]           
    • \[\therefore \,\,\]\[{{I}_{m}}=k-l\,\,{{I}_{m-1}}\Rightarrow k-l\,{{I}_{m-1}}=x{{(\log x)}^{m}}-m{{I}_{m-1}}\]                   
    • Þ \[k=x{{(\log x)}^{m}},l=m\].


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