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General Solution of Trigonometric Equation

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 $\int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin x] d x$ is $\frac{π}{a}$ where value of $a$ is

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Solution

$I = \int_{\frac{π}{2}}^{\frac{5 π}{6}} (1) d x + \int_{\frac{5 π}{6}}^{π} (0) d x + \int_{π}^{\frac{7 π}{6}} (- 1) d x + \int_{\frac{7 π}{6}}^{\frac{3 π}{2}} (- 2) d x$
$I = (\frac{5 π}{6} - \frac{π}{2}) + 0 - (\frac{7 π}{6} - π) - 2 (\frac{3 π}{2} - \frac{7 π}{6})$
$I = - \frac{π}{2} \Rightarrow a = - 2$

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