Definite Integration
Wallis / Beta — All Correct
Grade 12
Question:
<p>Identify all correct statements:</p>
<li>\(\displaystyle\int_0^{\pi/2}\sin^2 x\,dx = \dfrac{\pi}{4}\)</li>
<li>\(\displaystyle\int_0^{\pi/2}\cos^2 x\,dx = \dfrac{\pi}{4}\)</li>
<li>\(\displaystyle\int_0^{\pi/2}\sin^2 x\,dx = \int_0^{\pi/2}\cos^2 x\,dx\)</li>
<li>\(\displaystyle\int_0^{\pi/2}(\sin^n x+\cos^n x)\,dx = \int_0^{\pi/2}\sin^n x\,dx+\int_0^{\pi/2}\cos^n x\,dx\)</li>
Step-by-Step Solution
Key Concept: All four are correct. A and B follow from the half-angle formula. C follows from A=B. D is linearity of integration.
<div class='solution'>
<p><strong>A & B:</strong> $\int_0^{\pi/2}\sin^2 x\,dx = \int_0^{\pi/2}\frac{1-\cos 2x}{2}\,dx = \frac{\pi}{4}$. By King's ($x\to\pi/2-x$), $\cos^2 x$ integral equals same = $\frac{\pi}{4}$. ✓</p>
<p><strong>C:</strong> Follows immediately from A = B. ✓</p>
<p><strong>D:</strong> Linearity of the integral: always true. ✓</p>
<p>All four options are correct.</p>
</div>
Correct Answer: ['A', 'B', 'C', 'D']