JEE Main & Advanced JEE Main Paper (Held On 12 April 2014)

  • question_answer
    If [ ] denotes the greatest integer function, then the integral \[\int\limits_{0}^{\pi }{\left[ \cos x \right]}dx\]is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 )

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

    B) 0

    C) -1                                           

    D) \[-\frac{\pi }{2}\]

    Correct Answer: D

    Solution :

                    Let \[I=\int\limits_{0}^{\pi }{[\cos x]dx}\]                                             ?(1) \[I=\int\limits_{0}^{\pi }{[\cos (\pi -x)]dx}=\int\limits_{0}^{\pi }{[-\cos x]dx}\]   ?.(2) On adding (1) and (2), we get \[2I=\int\limits_{0}^{\pi }{[\cos x]dx+}\int\limits_{0}^{\pi }{[-\cos x]dx}\] \[2I=\int\limits_{0}^{\pi }{[\cos x]+}[-\cos x]dx\] \[2I=\int\limits_{0}^{\pi }{-1dx}(\because [x]+[-x]=-1\,if\,x\notin Z)\] \[\left. 2I=-x \right|_{0}^{\pi }=-\pi \]\[\Rightarrow \]\[I=\frac{-\pi }{2}\]

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