Question

The integral \[ \int_{0}^{\pi/2}\frac{\sin x}{x^p}\,dx \] converges if ____.

• A. \(p>2\)
• B. \(p>3\)
• C. \(p<2\)
• D. \(p>4\)

Hint:

Whenever an improper integral involves \[ \sin x,\ \tan x,\ 1-\cos x \] near \(x=0\), use \[ \boxed{ \sin x\sim x,\qquad \tan x\sim x,\qquad 1-\cos x\sim\frac{x^2}{2}. } \]

Answer 1

The Correct Option is C

Concept: The convergence depends on the behaviour of the integrand near \[ x=0. \] Using the standard approximation, \[ \boxed{ \sin x\sim x \qquad(x\to0) } \] Therefore, \[ \frac{\sin x}{x^p} \sim x^{\,1-p}. \]

Step 1: Apply the convergence criterion. The integral \[ \int_0 x^n\,dx \] converges if \[ n>-1. \] Here, \[ n=1-p. \] Hence, \[ 1-p>-1. \]

Step 2: Solve for \(p\). \[ 2>p \] or \[ \boxed{p<2.} \] Therefore, the correct option is \[ \boxed{(C)\;p<2.} \]