Problem 75 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int_{0}^{\pi}[\cot x] d x,[\cdot]\) denotes the greatest integer function, is equal to (A) \(\frac{\pi}{2}\) (B) 1 (C) \(-1\) (D) \(-\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Understanding the Problem

The given integral \( \int_{0}^{\pi} [\cot x] \, dx \) involves the greatest integer function, denoted by brackets \([\cdot]\). Our task is to evaluate it over the interval from 0 to \(\pi\). The \([\cot x]\) means we take the integer part (floor) of the \(\cot x\) function. We have to consider the behavior of \(\cot x\) between 0 and \(\pi\).

Step 2: Behavior of \(\cot x\) on the Interval

For \(x \in (0, \pi)\), \(\cot x\) approaches infinity as \(x\) approaches 0 from the right, decreases to 0 as \(x\) approaches \(\frac{\pi}{2}\), and moves towards negative infinity as \(x\) approaches \(\pi\) from the left. The function is undefined at \(x = 0\) and \(x = \pi\).

Step 3: Evaluating the Greatest Integer Function

For \(x \in (0, \frac{\pi}{2})\), \(\cot x > 1\) so \([\cot x] = 1\). For \(x = \frac{\pi}{2}\), \(\cot x = 0\), so \([\cot x] = 0\). For \(x \in (\frac{\pi}{2}, \pi)\), \(\cot x < 0\), leading \([\cot x] = -1\).

Step 4: Splitting the Integral

Split the integral at \(x = \frac{\pi}{2}\): \[\int_{0}^{\pi} [\cot x] \, dx = \int_{0}^{\frac{\pi}{2}} [\cot x] \, dx + \int_{\frac{\pi}{2}}^{\pi} [\cot x] \, dx.\]From previous evaluations: - \([\cot x] = 1\) for \(x \in (0, \frac{\pi}{2})\)- \([\cot x] = -1\) for \(x \in (\frac{\pi}{2}, \pi)\).

Step 5: Calculating Each Integral

Calculate each part: - For the first integral, \(\int_{0}^{\frac{\pi}{2}} 1 \, dx = 1 \times \frac{\pi}{2} = \frac{\pi}{2}\).- For the second integral, \(\int_{\frac{\pi}{2}}^{\pi} (-1) \, dx = -1 \times \left(\pi - \frac{\pi}{2}\right) = -1 \times \frac{\pi}{2} = -\frac{\pi}{2}\).

Step 6: Adding the Integrals

Finally, add the results of the two integrals:\[\frac{\pi}{2} + (-\frac{\pi}{2}) = 0.\]

Key Concepts

Greatest Integer Function

The greatest integer function, often represented with the notation \([\cdot]\), plays a crucial role in integral calculus when it comes to handling discrete changes of a continuous function. This function essentially takes any real number and "rounds it down" to the nearest integer, known as the floor value. For instance:

• For a value of 3.9, the greatest integer function gives 3.
• If you input -1.2, it outputs -2 since it rounds down negative values as well.

It's important to note that this operation is applied separately within any given interval when evaluating an integral. In the problem at hand with \([\cot x]\), it's essential to understand how cotangent values translate into specific integers over the interval \(0 \, \text{to} \, \pi\). When \(\cot x > 0\), \[\cot x\] takes on positive integer values, but as \([\cot x]\) transitions through zero to negative infinity, its greatest integer changes too. Understanding this behavior is key to breaking the integral into manageable parts.

Cotangent Function

The cotangent function, denoted as \(\cot x\), is a trigonometric function which is the reciprocal of the tangent function: \(\cot x = 1/\tan x\). It provides essential insights into angle relationships and measures in trigonometry.
- The cotangent function is particularly noteworthy between \(0 \, \text{and} \, \pi\) where it exhibits distinctive behavior:

• As \(x\) approaches 0, \(\cot x\) heads towards infinity.
• It decreases smoothly until \((\pi/2)\), where it becomes zero.
• Beyond \((\pi/2)\), \(\cot x\) shifts to negative values as it asymptotically approaches negative infinity near \(\pi\).

This oscillation between positive and negative values over a single period is distinctive of \(\cot x\) and is crucial when integrated over specific intervals as it determines the integer outputs of \[\cot x\], which vary accordingly. Integration of the cotangent thus often requires segmenting the integral based on these sign changes.

Definite Integration

Definite integration involves calculating the area under a function's curve between two limits. Unlike indefinite integration, which focuses on the antiderivative, definite integration provides a specific numeric result representing the accumulation of quantities.
The process starts by:

• Identifying integrable segments where the function behaves uniformly, especially important with functions like the step-wise \([\cot x]\) involved here.
• Computing the accumulated area by integrating over each identified segment, as seen in splitting the integral at \(x = \pi/2\).

For the specific integral \(\int_0^\pi [\cot x] \, dx\), the integral is split to manage different integer values of \[\cot x\] across sub-intervals. This ensures accuracy in calculating each segment's contribution to the total area, followed by summing these separate integrals to approach the final answer. Precision in segmenting and understanding integer transitions leads to correct evaluation, such as evidenced in arriving at the zero-sum result for this particular problem.