Definite Integration
Advanced Substitution
Grade 12
Question:
<p>Which is true about \(I=\displaystyle\int_0^{\pi/2}\frac{x}{\sin x}\,dx\)? [JEE Advanced 2015]</p>
\(I\) diverges
\(I=\dfrac{\pi\ln 2}{2}\)
\(I=\dfrac{\pi^2}{8}\)
\(I=\dfrac{\pi}{2}\)
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>
Correct Answer: B