Problem 162 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} d x\) lies between (A) \(\frac{\pi^{2}}{3}\) and \(\frac{2 \pi^{2}}{3}\) (B) \(\frac{\pi^{2}}{9}\) and \(\frac{2 \pi^{2}}{9}\) (C) \(\frac{2 \pi^{2}}{9}\) and \(\frac{4 \pi^{2}}{9}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these, as calculated bounds don't match any given options.

Step 1: Understand the integral

We need to evaluate the definite integral \( \int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} \, dx \). This integral doesn't have a standard elementary function as its solution, so we will need to estimate the value or bound it between known values.

Step 2: Estimate the bounds for integration

The function \( \frac{x}{\sin x} \) needs to be analyzed over the interval \( \left[ \frac{\pi}{6}, \frac{\pi}{2} \right] \). Note that in this range, \( \frac{x}{\sin x} \) is increasing, as \( x \) increases and \( \sin x \) increases.Consider the behavior:- At \( x = \frac{\pi}{6} \), \( \frac{\sin \pi/6}{\underline{\phantom{xx}}} = 0.5 \), so \( \frac{x}{\sin x} = \frac{\pi/6}{0.5} = \frac{\pi}{3} \).- As \( x \rightarrow \frac{\pi}{2} \), \( \sin x \rightarrow 1 \), so \( \frac{x}{\sin x} \approx x = \frac{\pi}{2} \).

Step 3: Use bounding to approximate the integral

Since \( \frac{x}{\sin x} \) increases from \( \frac{\pi}{3} \) to \( \frac{\pi}{2} \), we can bound the integral:\[ \int_{\pi/6}^{\pi/2} \frac{\pi}{3} \, dx \leq \int_{\pi/6}^{\pi/2} \frac{x}{\sin x} \, dx \leq \int_{\pi/6}^{\pi/2} \frac{\pi}{2} \, dx \]Calculating these bounds gives:\[ \frac{\pi}{3} \left( \frac{\pi}{2} - \frac{\pi}{6} \right) = \frac{\pi}{3} \cdot \frac{\pi}{3} = \frac{\pi^2}{9} \]\[ \frac{\pi}{2} \left( \frac{\pi}{2} - \frac{\pi}{6} \right) = \frac{\pi}{2} \cdot \frac{\pi}{3} = \frac{\pi^2}{6} \]

Step 4: Compare calculated bounds with options

From Step 3, the integral \( \int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} \, dx \) lies between \( \frac{\pi^2}{9} \) and \( \frac{\pi^2}{6} \). Comparing with the given options:- (A) \( \frac{\pi^{2}}{3} \) to \( \frac{2 \pi^{2}}{3} \) is incorrect- (B) \( \frac{\pi^{2}}{9} \) to \( \frac{2 \pi^{2}}{9} \) is incorrect- (C) \( \frac{2 \pi^{2}}{9} \) to \( \frac{4 \pi^{2}}{9} \) is incorrect- (D) None of these is correct based on our calculation.

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential when working with integrals involving expressions like \( \frac{x}{\sin x} \). Here, \( \sin x \) is one of these key functions and represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. It varies between -1 and 1 for all real numbers, which greatly affects how integrals involving \( \sin x \) behave. In the range \( x \in \left[ \frac{\pi}{6}, \frac{\pi}{2} \right] \), \( \sin x \) moves from 0.5 to 1. This gradual increase showcases why understanding the behavior of trigonometric functions over different intervals is crucial. This understanding helps to predict changes in the function \( \frac{x}{\sin x} \), as both \( x \) and \( \sin x \) influence the integral's value.
By recognizing these properties, one can progress with bounding techniques to estimate integrals more accurately. These properties ensure that while \( \sin x \) approaches its maximum, \( \frac{x}{\sin x} \) becomes predominantly influenced by \( x \).

Bounding Techniques

Bounding techniques are instrumental in evaluating integrals when direct computation is complex. In our example, the integral \( \int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} \, dx \) involved a function without a straightforward antiderivative. Hence, we applied bounding by identifying that \( \frac{x}{\sin x} \) is increasing on the interval \( \left[ \frac{\pi}{6}, \frac{\pi}{2} \right] \).
In this interval, the function's lower bound is \( \frac{\pi}{3} \), and its upper bound is \( \frac{\pi}{2} \). Therefore, instead of dealing directly with the function's complexity, we estimate the integral by integrating these bounds separately:

• Lower bound: \( \int_{\pi/6}^{\pi/2} \frac{\pi}{3} \, dx \)
• Upper bound: \( \int_{\pi/6}^{\pi/2} \frac{\pi}{2} \, dx \)

By doing this, we successfully narrow down the range in which the actual value of the integral falls, allowing for comparison with given answer options. These steps highlight the power of bounding techniques in simplifying complex integrals by framing them within comprehensible limits.

Riemann Integration

Riemann integration offers a way to understand how integration works as the sum of infinite slices of area under a curve. For the integral \( \int_{\pi / 6}^{\pi / 2} \frac{x}{\sin x} \, dx \), although direct Riemann sum calculations are challenging, the concept helps to comprehend how the integral approximates the total area below the curve between specific bounds.
To effectively use Riemann integration principles here, we consider smaller sub-intervals within \( \left[ \frac{\pi}{6}, \frac{\pi}{2} \right] \) and observe the behavior of \( \frac{x}{\sin x} \). Since the function is increasing, the left Riemann sums (lower estimates) give a smaller area, and the right Riemann sums (upper estimates) give a larger area. This approximates our integral to be somewhere between \( \frac{\pi^2}{9} \) and \( \frac{\pi^2}{6} \), demonstrating why our chosen bounds were effective.
By embracing Riemann's approach, students can visualize how integration sums contribute to solving definite integrals, even when expressed through crucial trigonometric relationships like \( \sin x \).