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Question

If [ ] denotes the greatest integer function then the integral $$\displaystyle \int_{0}^{\pi}[cos x]dx$$ is equal to :


A
π2
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B
0
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C
1
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D
π2
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Solution

The correct option is D $$-\dfrac {\pi}{2}$$
We know that for $$\cos x\in [0,\dfrac {\pi}{2}]\Rightarrow 1\ge \cos x \ge 0$$ and $$\cos x\in [\dfrac {\pi}{2},\pi]\Rightarrow 0\ge \cos x \ge -1$$

$$\displaystyle \int _{ 0 }^{ \pi  }{ [\cos { x]dx= }  } \displaystyle \int _{ 0 }^{ \tfrac { \pi  }{ 2 }  }{ 0dx\quad + } \displaystyle \int _{ \tfrac { \pi  }{ 2 }  }^{ \pi  }{ -1dx}$$

$$=-\dfrac { \pi  }{ 2 }   $$  

Mathematics

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