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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
    The area bounded by the curves \[y=\ln x\], \[y=\ln |x|\], \[y=\,|\ln x|\] and \[y=\,|\ln |x||\] is [AIEEE 2002]

    A) 4 sq. unit                                

    B) 6 sq. unit

    C) 10 sq. unit                             

    D) None of these

    Correct Answer: A

    Solution :

    • We know that \[\log x\] is defined for \[x>0\] and \[\log |x|\] is defined for all \[x\in R-\{0\}\]           
    • Also \[|\log x|\ge 0\] and \[|\log \,|x|\,|\,\ge \,0\]           
    • \[\therefore \]Required area is symmetrical in all the four quadrants and is equal to \[=4\int_{0}^{1}{|\log x|dx=-4\int_{0}^{1}{\log x\,\,dx}}\], (\[\because \]In
    • \[(0,1),\log x<0)\]   =\[-4\,[x\log x-x]_{0}^{1}=-4(-1)=4\]sq.unit,\[\left( \because \,\,\,\underset{x\to 0}{\mathop{\lim }}\,x\log x=0 \right)\].


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