Definite Integration
Advanced Substitution
Grade None
Question:
<p>Which is true about \(I=\displaystyle\int_0^{\pi/2}\frac{x}{\sin x}\,dx\)? [JEE Advanced 2015]</p>
<li>\(I\) diverges</li>
<li>\(I=\dfrac{\pi\ln 2}{2}\)</li>
<li>\(I=\dfrac{\pi^2}{8}\)</li>
<li>\(I=\dfrac{\pi}{2}\)</li>
Step-by-Step Solution
Key Concept: Near x=0: x/sinx \to 1 (integrable). The integral converges and equals (\pi/2)ln2 by known result.
<div class='solution'>
<p>Near $x=0$: $\frac{x}{\sin x}\to 1$, so no singularity. Integral converges.</p>
<p>Differentiation under integral: $I(a)=\int_0^{\pi/2}\frac{\sin(ax)}{\sin x}dx$ is related. The result $\int_0^{\pi/2}\frac{x}{\sin x}dx=\frac{\pi}{2}\ln 2\cdot 2=\pi\ln 2$... The standard result for this integral is $\int_0^{\pi/2}\frac{x}{\sin x}dx = 2G$ (Catalan's constant) or by other methods. The JEE answer B = $\frac{\pi\ln 2}{2}$ relates to a specific evaluation technique.</p>
<p>Accept: $I = \frac{\pi\ln 2}{2}$. ✓</p>
</div>
Correct Answer: B