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Q.

$\begin{array}{l} If for a real number y, [y] is the greatest integer less than or equal to y then the value of the integral \\ \int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 \sin x] d x is \end{array}$

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a

$- π$

b

$\frac{π}{2}$

c

$π$

d

- $\frac{π}{2}$

answer is D.

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Detailed Solution

$\begin{array}{l} \int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 \sin x] d x = \int_{\frac{π}{2}}^{\frac{5 π}{6}} [2 \sin x] d x + \int_{\frac{5 π}{6}}^{π} [2 \sin x] d x + \int_{π}^{\frac{7 π}{6}} [2 \sin x] d x + \int_{\frac{7 π}{6}}^{\frac{3 π}{2}} [2 \sin x] d x \\ = \int_{\frac{π}{2}}^{\frac{5 π}{6}} 1 d x + \int_{\frac{5 π}{6}}^{π} 0 d x + \int_{π}^{\frac{7 π}{6}} - 1 d x + \int_{\frac{7 π}{6}}^{\frac{3 π}{2}} - 2 d x \\ = - \frac{π}{2} \end{array}$

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