Definite Integration Question 475
Question: The value of $ \int _{\pi }^{2\pi }{
[2\sin x]dx,} $ where $ [.] $ represents the greatest integer function, is [IIT 1995]
Options:
A) $ -\pi $
B) $ -2\pi $
C) $ -\frac{5\pi }{3} $
D) $ \frac{5\pi }{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
$ \int _{\pi }^{2\pi }{[2\sin x]dx=\int _{\pi }^{\pi +(\pi /6)}{(-1)dx+\int _{\pi +(\pi /6)}^{\pi +(\pi /2)}{(-2)dx}}} $
$ +\int _{\pi +(\pi /2)}^{\pi +(\pi /2)+(\pi /3)}{(-2)dx+\int _{\pi +(\pi /2)+(\pi /3)}^{2\pi }{(-1)dx}} $
$ =-\frac{\pi }{6}-2[ \frac{\pi }{2}-\frac{\pi }{6} ]-2[ \frac{\pi }{3} ]-1[ \frac{\pi }{2}-\frac{\pi }{3} ] $
$ =-\frac{\pi }{6}-\frac{2\pi }{3}-\frac{2\pi }{3}-\frac{\pi }{6} $
$ =-\frac{\pi }{6}-\frac{8\pi }{6}-\frac{\pi }{6}=-\frac{10\pi }{6}=-\frac{5\pi }{3} $ .
BETA
