KVPY Sample Paper KVPY Stream-SX Model Paper-25

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

    A) \[\pi \]

    B) \[\frac{\pi }{2}\]

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

    D) none of these

    Correct Answer: A

    Solution :

    let \[I=\int\limits_{{{\tan }^{-1}}\alpha }^{{{\cot }^{-1}}\alpha }{\frac{\tan x}{\tan x+\cot x}dx}\]\[=\int\limits_{{{\tan }^{-1}}\alpha }^{{{\cot }^{-1}}\alpha }{\frac{\cot x}{\cot x+\tan x}dx}\]     \[\left( \because {{\tan }^{-1}}\alpha +{{\cot }^{-1}}\alpha =\frac{\pi }{2} \right)\]
    \[\therefore \]      \[2I=\int\limits_{{{\tan }^{-1}}\alpha }^{{{\cot }^{-1}}\alpha }{dx={{\cot }^{-1}}\alpha -{{\tan }^{-1}}\alpha }\]          \[\Rightarrow \]   \[I=\frac{1}{2}\left[ \frac{\pi }{2}-2{{\tan }^{-1}}\alpha  \right]\]
    \[\therefore \]      \[-\frac{\pi }{2}<I<\frac{3\pi }{4}\forall \alpha \]

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