<p>The value of \(\displaystyle\int_0^\infty\frac{\sin x}{x}\,dx\) is [JEE Advanced 2006]</p>
Step-by-Step Solution
Key Concept: Classic Dirichlet integral = \pi/2. Via Feynman: I(a)=\int_0^\inftye^(-ax)sinx/x dx, I'(a)=-1/(1+a^2), I(\infty)=0, I(0)=\pi/2.
<div class='solution'>
<p>This is the Dirichlet integral: $\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2}$.</p>
<p><strong>Feynman method:</strong> Let $I(a)=\int_0^\infty\frac{e^{-ax}\sin x}{x}dx$.</p>
<p>$I'(a)=-\int_0^\infty e^{-ax}\sin x\,dx=-\frac{1}{a^2+1}$.</p>
<p>$I(a)=-\arctan a+C$. $I(\infty)=0\Rightarrow C=\pi/2$.</p>
<p>$I(0)=\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2}$. ✓</p>
</div>
Correct Answer: A