Mathematics

Value of  $$\int_{\pi}^{2 \pi} [2 \, sin \, x] dx $$where [ ] represents the greatest integer function is 


ANSWER

$$\frac {5 \pi }{3}$$


SOLUTION
$$\begin{matrix} let\, \, I=\int _{ \pi  }^{ 2\pi  }{ \left[ { 2\sin  x } \right]  } dx \\ =-\left[ { 1\left( { \frac { { 7\pi  } }{ 6 } -\pi  } \right) +2\left( { \frac { { 11\pi  } }{ 6 } +\frac { { 7\pi  } }{ 6 }  } \right) +i\left( { 2\pi -\frac { { 11\pi  } }{ 6 }  } \right)  } \right]  \\ =\frac { { -5\pi  } }{ 6 }  \\  \end{matrix}$$
$$\begin{matrix} for\, \, \, \pi \leqslant x\leqslant 2\pi  \\ \, \, \, \, \, \, \, \, -2\leqslant 2\sin  x\leqslant 0 \\ when\, \, \, \, 2\sin  =-1 \\ \, \, \, \, \, \, \, \, \, \, \, \Rightarrow \sin  x=-\frac { 1 }{ 2 }  \\ \, \, \, \therefore x=\frac { { 7\pi  } }{ 6 } ,\frac { { 11\pi  } }{ 6 }  \\ \, \, if\, \, -2\leqslant 2\sin  x<-1 \\ \, \, \, \, \, \Rightarrow \frac { { 7\pi  } }{ 6 } <x<\frac { { 11\pi  } }{ 6 }  \\  \end{matrix}$$
$$\begin{matrix} Now,\, I=\int { \frac { { A\left( { 6x+5 } \right) +B } }{ { 3{ x^{ 2 } }+5x+2 } } dx }  \\ \, \, \, \, \, \, \Rightarrow I=\int { \frac { { \frac { 5 }{ 6 } \left( { 6x+5 } \right) -\frac { 1 }{ 6 }  } }{ { 3{ x^{ 2 } }+5x+2 } } dx }  \\ \, \, \, \, \, \, \Rightarrow I=\frac { 5 }{ 6 } \int { \frac { { 6x+5 } }{ { 3{ x^{ 2 } }+5x+2 } } dx-\frac { 1 }{ 6 } \int { \frac { 1 }{ { 3{ x^{ 2 } }+5x+2 } } dx }  }  \\ \, \, \, \, \, \Rightarrow I=\frac { 5 }{ 6 } \log  \left| { 3{ x^{ 2 } }+5x+2 } \right| -\frac { 1 }{ 6 } \, \, { I_{ 1 } }\_ \_ \_ \_ \left( 1 \right)  \\  \end{matrix}$$
Where,
$$\begin{matrix} \, \, { I_{ 1 } }=\int { \frac { 1 }{ { 3{ x^{ 2 } }+5x+2 } } dx }  \\ \, \, \, \, \, \, \, =\int { \frac { 1 }{ { 3\left( { { x^{ 2 } }+\frac { 5 }{ 3 } x+\frac { 2 }{ 3 }  } \right)  } } dx }  \\ \, \, \, \, \, \, =\frac { 1 }{ 3 } \int { \frac { 1 }{ { { x^{ 2 } }+2\times x\times \frac { 5 }{ 6 } +{ { \left( { \frac { 5 }{ 6 }  } \right)  }^{ 2 } }-{ { \left( { \frac { 5 }{ 6 }  } \right)  }^{ 2 } }+\frac { 2 }{ 3 }  } } dx }  \\ \, \, \, \, \, =\frac { 1 }{ 3 } \int { \frac { { 1dx } }{ { { { \left( { x+\frac { 5 }{ 6 }  } \right)  }^{ 2 } }+{ { \left( { \frac { 5 }{ 6 }  } \right)  }^{ 2 } }+\frac { 2 }{ 3 }  } }  } \, \,  \\  \end{matrix}$$
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Single Correct Medium Published on 17th 09, 2020
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