Definite Integration
Grade 12
Question:
<p><span class="math-tex">\(\int\limits_0^{\frac{\pi }{2}} {[\sin x + \cos x]} \)</span>dx, where [ ] represents the greatest integer function, equals</p>
<p style="display:inline"><span class="math-tex">\(\frac {3\pi}{8}\)</span></p>
<p style="display:inline">0</p>
<p style="display:inline"><span class="math-tex">\(\frac {\pi}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac {\pi}{4}\)</span></p>
Step-by-Step Solution
<p>sin x <span class="math-tex">$\geq$</span> sin<sup>2</sup>x and cos x <span class="math-tex">$\geq$</span> cos<sup>2</sup> x in <span class="math-tex">$\left( {0,\frac{\pi }{2}} \right)$</span><br />
<span class="math-tex">$\left.\begin{array}{l} \Rightarrow \sin x+\cos x \geq 1 \\ \text { Also, } \sin x+\cos x \leq \sqrt{2} \end{array}\right\}$</span> <span class="math-tex">$\Rightarrow$</span> [sin x + cos x] = 1<br />
<span class="math-tex">$\int \limits_{0}^{\frac{\pi}{2}}[\sin x+\cos x]$</span>dx = <span class="math-tex">$\int \limits_{0}^{\frac{\pi}{2}} d x=\frac{\pi}{2}$</span></p>
Correct Answer: C