If $[a]$ denote the greatest integer which is less than or equal to $a$ .
Then, the value of the integral $\int_{- \frac{π}{2}}^{\frac{π}{2}} [sin x cos x] d x$ is?

\frac{π}{2}

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π

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- π

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- \frac{π}{2}

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Solution

The correct option is C $- \frac{π}{2}$
Let $I = \int_{- \frac{π}{2}}^{\frac{π}{2}} [sin x cos x] d x$
$= \int_{- \frac{π}{2}}^{\frac{π}{2}} [\frac{1}{2} sin 2 x] d x$
Put $θ = 2 x$ $\Rightarrow$ $d θ 2 d x$
Also when $x = - π / 2$ , then $θ = - π$
when $x = \frac{π}{2}$ , then $θ = π$
Then, $I = \frac{1}{2} \int_{- π}^{π} [\frac{1}{2} sin θ] d θ$
$= \frac{1}{2} [\int_{- π}^{0} (- 1) d x + \int_{0}^{π} (0) d x]$
$= \frac{1}{2} {[- x]}_{- π}^{0} + 0 = - \frac{π}{2}$

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