Definite Integration
Reduction + Trig-Multi
Grade 12
Question:
<p>For all \(n\in\mathbb{N}\), \(\int_0^{\pi/2}\sin^n x\,dx\) compared to \(\int_0^{\pi/2}\cos^n x\,dx\): [JEE Advanced 2010]</p>
<li>Equal</li>
<li>\(\sin^n\) integral \(>\) \(\cos^n\) integral</li>
<li>\(\sin^n < \cos^n\)</li>
<li>Cannot determine</li>
Step-by-Step Solution
Key Concept: King's rule (x\to \pi/2-x): \int_0^(\pi/2) sinⁿx dx = \int_0^(\pi/2) cosⁿx dx. They are always equal.
<div class='solution'>
<p>Let $I=\int_0^{\pi/2}\sin^n x\,dx$. King ($x\to\pi/2-x$): $\sin(\pi/2-x)=\cos x$.</p>
<p>$$I=\int_0^{\pi/2}\cos^n x\,dx$$</p>
<p>They are equal for <em>all</em> $n$. ✓</p>
</div>
Correct Answer: A