JEE Main & Advanced Sample Paper JEE Main Sample Paper-47

  • question_answer
    The integral \[\int_{{{\tan }^{-1}}\lambda }^{{{\cot }^{-1}}\lambda }{\frac{\tan x}{\tan x+\cot x}dx,}\]\[\forall \lambda \in R\] cannot take the value

    A)  \[\frac{-\pi }{4}\]                              

    B) \[\frac{\pi }{4}\]

    C)  \[\frac{+\pi }{2}\]                            

    D)  \[\frac{-3\pi }{4}\]

    Correct Answer: A

    Solution :

     Let \[I=\,\int_{{{\tan }^{-1}}}^{{{\cot }^{-1}}}{\frac{\tan \,x}{\tan \,x+\cot \,x}dx}\]                      ?(i) \[I=\int_{{{\tan }^{-1}}\lambda }^{{{\cot }^{-1}}\lambda }{\frac{\cot \,x}{\cot \,x+\tan \,x}dx}\]    ?(ii) \[\left( \because \,\,{{\tan }^{-1}}x+{{\cot }^{-1}}x=\frac{\pi }{2} \right)\] On adding Eqs. (i) and (ii), we get \[2I=\int_{{{\tan }^{-1}}\lambda }^{{{\cot }^{-1}}\lambda }{1\,dx}\,\,\,\,=[x]_{{{\tan }^{-1}}\lambda }^{{{\cot }^{-1}}\lambda }\] \[\Rightarrow \]            \[2I={{\cot }^{-1}}\,\lambda -{{\tan }^{-1}}\lambda \] \[\Rightarrow \]            \[2I=\frac{\pi }{2}\,-{{\tan }^{-1}}\lambda -{{\tan }^{-1}}\lambda \] \[\Rightarrow \]            \[I=\frac{\pi }{4}-{{\tan }^{-1}}\lambda \] \[\Rightarrow \]            \[\frac{-\pi }{2}<{{\tan }^{-1}}\lambda <\frac{\pi }{2}\] \[\frac{3\pi }{4}>\frac{\pi }{4}-{{\tan }^{-1}}\,\lambda >-\frac{\pi }{4}\] \[\Rightarrow \]            \[\frac{-\pi }{4}<I<\frac{3\pi }{4}\]

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