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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    \[\int_{{}}^{{}}{{{(\log x)}^{2}}\ dx=}\]  [IIT 1971, 77]

    A) \[x{{(\log x)}^{2}}-2x\log x-2x+c\]

    B) \[x{{(\log x)}^{2}}-2x\log x-x+c\]

    C) \[x{{(\log x)}^{2}}-2x\log x+2x+c\]        

    D) \[x{{(\log x)}^{2}}-2x\log x+x+c\]

    Correct Answer: C

    Solution :

    • \[\int_{{}}^{{}}{{{(\log x)}^{2}}dx}\]. Put \[\log x=t\Rightarrow {{e}^{t}}=x\Rightarrow dx={{e}^{t}}dt,\] then it reduces to \[\int_{{}}^{{}}{{{t}^{2}}.\,{{e}^{t}}dt={{t}^{2}}{{e}^{t}}-2t{{e}^{t}}+2{{e}^{t}}+c}\]                                                            
    • \[=x{{(\log x)}^{2}}-2x\log x+2x+c\].


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