2. Questions Bank
  3. 12th Class
  4. Mathematics
  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If for a real number \[y,\,\,[y]\] is the greatest integer less than or equal to \[y,\] then the value of the integral \[\int\limits_{\pi /2}^{3\pi /2}{[2\sin x]\,dx}\] is [IIT 1999]

    A) \[-\pi \]                                  

    B) 0

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

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

    Correct Answer: C

    Solution :

    • We know \[-1\le \sin x\le 1\Rightarrow -2\le 2\sin x\le 2\]                   
    • \[I=\int_{\frac{\pi }{2}}^{\frac{3\pi }{2}}{[2\sin x]dx}\]\[=\int_{\frac{\pi }{2}}^{\frac{5\pi }{6}}{[2\sin x]dx+\int_{\frac{5\pi }{6}}^{\pi }{[2\sin x]dx}}\] \[+\int_{\pi }^{\frac{7\pi }{6}}{[2\sin x]dx+\int_{\frac{7\pi }{6}}^{\frac{3\pi }{2}}{[2\sin x]dx}}\]                   
    • \[=\int_{\frac{\pi }{2}}^{\frac{5\pi }{6}}{(1)dx+\int_{\frac{5\pi }{6}}^{\pi }{(0)dx+\int_{\pi }^{\frac{7\pi }{6}}{(-1)dx+\int_{\frac{7\pi }{6}}^{\frac{3\pi }{2}}{(-2)}}dx}}\]                   
    • \[=\left( \frac{5\pi }{6}-\frac{\pi }{2} \right)+0-\left( \frac{7\pi }{6}-\pi  \right)-2\left( \frac{3\pi }{2}-\frac{7\pi }{6} \right)\]                    \[=\frac{2\pi }{6}-\frac{\pi }{6}-\frac{4\pi }{6}=-\frac{\pi }{2}\].


You need to login to perform this action.
You will be redirected in 3 sec spinner