Problem 214 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int_{0}^{\pi}[\cot x] d x,[.]\) 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 is \(-\frac{\pi}{2}\). Answer: (D) \(-\frac{\pi}{2}\).

Step 1: Understand the Function Inside the Integral

The function to integrate is \([\cot x]\), where \([.]\) denotes the greatest integer function, also known as the floor function. For each \(x\), \([\cot(x)]\) is the greatest integer less than or equal to \(\cot(x)\).

Step 2: Analyze the Cotangent Function in Interval

The cotangent function \(\cot(x)\) is defined as \(\frac{\cos(x)}{\sin(x)}\). It is decreasing and undefined at multiples of \(\pi\), i.e., \(x=n\pi\) for integers \(n\). Within the interval \( (0, \pi) \), \(\cot(x)\) starts from large values near \(x = 0\) and decreases to \(-\infty\) as \(x\) approaches \(\pi\) from the left.

Step 3: Identify Important Intervals

To determine the behavior of \([\cot(x)]\) in \([0, \pi]\), identify key intervals caused by changes in integer values of \(\cot(x)\):- Near \(x \to 0^+\), \(\cot(x) \to \infty\)- \(x = \frac{\pi}{4}\), \(\cot(x) = 1\), so \([\cot(x)] = 1\)- \(x = \frac{\pi}{2}\), \(\cot(x) = 0\), so \([\cot(x)] = 0\)- Near \(x = \frac{3\pi}{4}\), \(\cot(x) = -1\), so \([\cot(x)] = -1\)

Step 4: Determine Values in each sub-interval

In the interval \((0, \frac{\pi}{2})\), \([\cot(x)] = 0\).In \((\frac{\pi}{2}, \pi)\), \(\cot(x) < 0\) and starts decreasing from \(-\infty\), causing \([\cot(x)]\) to be \(-1\) throughout this interval.

Step 5: Calculate the Integral of Each Interval

Now, compute the integral for each interval:- From \([0, \frac{\pi}{2}]\): \[ \int_0^{\frac{\pi}{2}} 0 \, dx = 0 \] - From \((\frac{\pi}{2}, \pi)\): \[ \int_{\frac{\pi}{2}}^{\pi} -1 \, dx = -1 \times \left(\pi - \frac{\pi}{2}\right) = -\frac{\pi}{2} \]

Step 6: Combine Results

Sum the integrals for complete interval:\[ 0 + (-\frac{\pi}{2}) = -\frac{\pi}{2} \]

Step 7: Choose the Correct Answer

The computed integral from \(0\) to \(\pi\) of \([\cot x]\) is \(-\frac{\pi}{2}\). Thus, the correct answer is (D) \(-\frac{\pi}{2}\).

Key Concepts

Greatest Integer Function

The greatest integer function, often referred to as the floor function, is a mathematical function that rounds down a real number to the nearest integer less than or equal to it. In mathematical notation, it is symbolized as \([x]\). For any real number \(x\), \([x]\) represents the largest integer that is not greater than \(x\).
For example:

• \([3.7] = 3\)
• \([-2.1] = -3\)

This function is particularly useful in cases where the need arises to quantify the size of non-integer values in terms of whole numbers. Within the context of calculus, when combined with various functions such as trigonometric functions, it sometimes alters from the smooth behavior of a line or curve to a step-like appearance. This discrete nature highlights integer levels and plays a critical role in certain definite integral calculations.

Cotangent Function

The cotangent function is one of the basic trigonometric functions. It is the reciprocal of the tangent function. In terms of sine and cosine, the cotangent function is expressed as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Cotangent, along with other trigonometric functions, operates within periodic intervals, repeating its values over specific intervals.
Key characteristics of the cotangent function include:

• It is undefined at integer multiples of \(\pi\), such as \(x = \pi, 2\pi, \ldots\) because \(\sin(x) = 0\) at these points.
• Within each interval \((0, \pi), (\pi, 2\pi), \ldots\), the function decreases, starting from positive infinity and approaching negative infinity.

In the interval \((0, \pi)\) specifically, cotangent has a unique behavior where it decreases continuously, capturing a broad range of values, which later simplifies integration by the greatest integer function application in definite integrals.

Definite Integral

A definite integral represents the accumulation of quantities, such as areas under curves, between defined limits. Given by the notation \(\int_{a}^{b} f(x)\, dx\), the process involves finding the integral of a function \(f(x)\) from \(x = a\) to \(x = b\). This fundamental concept in integral calculus provides a numerical value rather than an indefinite expression.
For definite integrals:

• The limits \(a\) and \(b\) can define specific sections of a curve, providing the complete positive and negative accumulations between these bounds.
• The result can be either zero, positive, or negative, denoting the area under the curve and above the x-axis between the specified limits.

In the context of our exercise, the integral \(\int_{0}^{\pi}[\cot x] \, dx\) reflects numerical integration, considering the step-like pattern of the floor function applied to \(\cot(x)\). Splitting the integration into recognizable intervals and understanding the behavior of \([\cot(x)]\) within \((0, \pi)\) ensures the accurate calculation of areas, which in turn confirms that the integral sums up to the value \(-\frac{\pi}{2}\).