Definite Integration
Parseval / Orthogonality
Grade 12
Question:
<p>For integers \(m,n\), \(\displaystyle\int_0^{2\pi}\sin(mx)\cos(nx)\,dx = ?\) [JEE Advanced 2007]</p>
Step-by-Step Solution
Key Concept: sin(mx)cos(nx) = (1/2)[sin((m+n)x)+sin((m-n)x)]. Integral of sin(kx) over full period = 0 for any integer k (including k=0: sin(0)=0).
<div class='solution'>
<p>$\sin(mx)\cos(nx)=\frac{1}{2}[\sin((m+n)x)+\sin((m-n)x)]$</p>
<p>$$\int_0^{2\pi}\sin(kx)\,dx = \begin{cases}0&k\ne 0\\0&k=0\end{cases}$$</p>
<p>So $\int_0^{2\pi}\sin(mx)\cos(nx)\,dx=0$ for all integers $m,n$. ✓</p>
Correct Answer: A