JEE Main & Advanced JEE Main Paper (Held On 22 April 2013)

  • question_answer
    The integral \[\int\limits_{7\pi /4}^{7\pi /3}{\sqrt{{{\tan }^{2}}}x\operatorname{d}x}\] is equal to :     JEE Main  Online Paper (Held On 22 April 2013)

    A)  \[\log 2\sqrt{2}\]                

    B)  \[\log 2\]

    C)  \[2\log 2\]                           

    D)  \[\log \sqrt{2}\]

    Correct Answer: D

    Solution :

     Let \[I=\int\limits_{7\pi /4}^{7\pi /3}{\sqrt{{{\tan }^{2}}x\,dx}}\] \[=\int\limits_{7\pi /4}^{7\pi /3}{\tan x\,dx}=-\log \,cos\,\left. x \right|_{7\pi /4}^{7\pi /3}\] \[=-\left[ \log \,\cos \frac{7\pi }{3}-\log \cos \frac{7\pi }{4} \right]\] \[=\log \,cos\,\frac{7\pi }{4}-\log \,cos\,\frac{7\pi }{3}\] \[=\log \left[ \frac{\cos \frac{7\pi }{4}}{\cos \frac{7\pi }{3}} \right]=\log \left[ \frac{\cos \left( 2\pi -\frac{\pi }{4} \right)}{\cos \left( 2\pi +\frac{\pi }{3} \right)} \right]\] \[=\log \left( \frac{\cos \frac{\pi }{4}}{\cos \frac{\pi }{3}} \right)=\log \left( \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} \right)\] \[=\log \left( \frac{2}{\sqrt{2}} \right)=\log \sqrt{2}.\]

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