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JEE Main & Advanced Mathematics Definite Integration Question Bank Fundamental definite integration, Definite integration by substitution

  • question_answer
    If \[g(1)=g(2)\], then \[\int_{1}^{2}{{{\left[ fg(x) \right]}^{-1}}}f'\{g(x)\}\ g'(x)\ dx\]is equal to                                                 [AMU 2005]

    A)                 1             

    B)                 2

    C)                 0             

    D)                 None of these

    Correct Answer: C

    Solution :

               \[I=\int_{1}^{2}{{{[f\{g(x)\}]}^{-1}}f'[g(x)]\,g'(x)\ dx}\]            Let \[f\{g(x)\}=z\] Þ \[f'\{g(x)\}\ g'(x)\ dx=dz\]            When \[x=1,\ z=f\ \{g\ (1)\}\]            When \[x=2,\ z=f\ \{g\ (2)\}\]            \[\therefore \] \[I=\int_{f\{g(1)\}}^{f\{g(2)\}}{\frac{1}{z}dz}=|\log z|_{f\{g(1)\}}^{f\{g(2)\}}\]                                 \[\Rightarrow \]I\[=\log f\ \{g(2)\}-\log f\ \{g(1)\}=0\], (\[\because \] g(2)=g(1)).


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