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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    If \[\int_{{}}^{{}}{\frac{2x+3}{{{x}^{2}}-5x+6}}\ dx=9\ \ln (x-3)-7\ln (x-2)+A\], then \[A=\]                [MP PET 1992]

    A) \[5\ln (x-2)+\]Constant

    B) \[-4\ln (x-3)+\]constant

    C) Constant

    D) None of these

    Correct Answer: C

    Solution :

    • \[\int_{{}}^{{}}{\frac{2x+3}{{{x}^{2}}-5x+6}dx=\int_{{}}^{{}}{\frac{2x-5}{{{x}^{2}}-5x+6}dx+\int_{{}}^{{}}{\frac{8}{{{x}^{2}}-5x+6}}\ dx}}\]                   
    • \[=\frac{-\cos \frac{x}{8}}{\left( \frac{1}{8} \right)}+\frac{\sin \frac{x}{8}}{\left( \frac{1}{8} \right)}+c\].                   
    • \[=\log [(x-2)(x-3)]+8\int_{{}}^{{}}{\left[ \frac{1}{x-3}-\frac{1}{x-2} \right]dx+c}\].                   
    • \[=\log (x-2)+\log (x-3)+8\log (x-3)-8\log (x-2)+c\]                   
    • \[=9\log (x-3)-7\log (x-2)+c\]                   .....(i)                   
    • Now given that                   
    • \[\int_{{}}^{{}}{\frac{2x+3}{{{x}^{2}}-5x+6}}\text{  }dx=9\log (x-3)-7\log (x-2)+A\]                
    • Equating it to (i), we get \[A=\]constant.


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