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Integration of Piecewise Continuous Functions

If for a real...

Question

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

- π

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0

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- \frac{π}{2}

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\frac{π}{2}

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Solution

The correct option is C $- \frac{π}{2}$
$I = \frac{5 π}{6} \int \frac{π}{2} (1) d x + π \int \frac{5 π}{6} 0. d x + \frac{7 π}{6} \int π (- 1) d x + \frac{3 π}{2} \int \frac{7 π}{6} (- 2) d x$
$= (\frac{5 π}{6} - \frac{π}{2}) + 0 - (\frac{7 π}{6} - π) - 2 (\frac{3 π}{2} - \frac{7 π}{6})$
$I = - \frac{π}{2}$

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