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Standard Limits to Remove Indeterminate Form

lf for any re...

Question

lf for any real value $y$ , $[y] =$ the greatest integer less than or equal to $x$ then the value of $\int_{\frac{π}{2}}^{\frac{3 x}{2}} [2 sin x] d x$ equals

\frac{π}{2}

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- \frac{π}{2}

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- π

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0

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Solution

The correct option is B $- \frac{π}{2}$
$\int_{\frac{π}{2}}^{\frac{3 π}{2}} [2 sin x] d x = \int_{\frac{π}{2}}^{\frac{5 π}{6}} [2 sin x] d x$
$+ \int_{\frac{5 π}{6}}^{\frac{7 π}{6}} [2 sin x] d x + \int_{\frac{7 π}{6}}^{\frac{3 π}{2}} [2 sin x] d x$
$= \int_{\frac{π}{2}}^{\frac{5 π}{6}} 0 d x + \int_{\frac{5 π}{6}}^{\frac{7 π}{6}} (- 1) d x + \int_{\frac{7 π}{6}}^{\frac{3 π}{2}} (- 2) d x$
$= {[- x]}_{\frac{5 π}{6}}^{\frac{7 π}{6}} + {[- 2 x]}_{\frac{7 π}{6}}^{\frac{3 π}{2}} = - \frac{π}{2}$

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