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

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 evaluates to \(-\frac{\pi}{2}\), which is option (D).

Step 1: Understanding the Integral

The goal is to evaluate the integral \( \int_{0}^{\pi} [\cot x] \, dx \), where \([ \cdot ]\) denotes the greatest integer function (also known as the floor function). We need to find where \( \cot x \) takes integer values in the interval \( [0, \pi] \).

Step 2: Analyzing the Behavior of \( \cot x \)

The function \( \cot x \) transitions from \(+\infty\) to \(-\infty\) as \(x\) ranges from 0 to \(\pi\). Since \(\cot \frac{\pi}{2} = 0\), and the behavior is symmetric around \(\frac{\pi}{2}\), \( \cot x \) changes its integer part at \(x = \frac{\pi}{4}, \frac{3\pi}{4}\).

Step 3: Splitting the Integral

The integral can be split at points where \( \cot x \) crosses integer boundaries. Hence, the split is at points where \( \cot x \) crosses integer values: from 0 to \( \frac{\pi}{4} \) where \( [\cot x] = 0 \); then from \( \frac{\pi}{4} \) to \( \frac{3\pi}{4} \) where \( [\cot x] = -1 \); and from \( \frac{3\pi}{4} \) to \( \pi \) where \( [\cot x] = 0 \).

Step 4: Evaluating Each Subinterval

On each interval, evaluate the integral: 1. From \(0\) to \(\frac{\pi}{4}\), \([\cot x] = 0\), integral is \(\int_{0}^{\frac{\pi}{4}} 0 \, dx = 0\) 2. From \(\frac{\pi}{4}\) to \(\frac{3\pi}{4}\), \([\cot x] = -1\), integral is \(\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} -1 \, dx = -\frac{\pi}{2}\) 3. From \(\frac{3\pi}{4}\) to \(\pi\), \([\cot x] = 0\), integral is \(\int_{\frac{3\pi}{4}}^{\pi} 0 \, dx = 0\).

Step 5: Combining Results

Sum the results of each subinterval: \[ 0 + (-\frac{\pi}{2}) + 0 = -\frac{\pi}{2} \]

Step 6: Conclusion

Thus, the value of the integral \( \int_{0}^{\pi} [\cot x] \, dx \) is \(-\frac{\pi}{2}\). Therefore, the correct answer is option (D).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, serving as a numerical value representing the area under a curve between two points on the x-axis. In mathematical terms, when we calculate a definite integral, we sum up infinite infinitesimal quantities along an interval. This operation allows us to find the accumulated total of a rate of change, such as distance, area, or volume.

Definite integrals are defined as follows:

• The symbol \(\int_{a}^{b} f(x) \, dx\) represents the integral of \(f(x)\) from \(a\) to \(b\).
• \(a\) and \(b\) are the limits of integration, defining where to start and stop along the x-axis.
• The function \(f(x)\) is the integrand, the function to be integrated.

In the exercise provided, the aim is to solve \(\int_{0}^{\pi} [\cot x] \, dx\). The function \(\cot x\) is the integrand within the limits from 0 to \(\pi\). By evaluating the definite integral, we determine the net area under the function \([\cot x]\) over this specified interval.

Greatest Integer Function

The greatest integer function, often represented as \([x]\) or sometimes \(\lfloor x \rfloor\), is essential in calculus, particularly in dealing with piecewise functions. This function returns the largest integer less than or equal to \(x\). In other terms, it "rounds down" a given number to the nearest integer.

Some key properties include:

• If \(x\) is already an integer, \([x] = x\).
• For every non-integer \(x\), \([x] = n\) where \(n\) is an integer such that \(n \leq x < n+1\).

The greatest integer function is particularly useful in managing continuous functions and converting them to discrete stepping functions, which is often useful in real-world applications such as digital signal processing. In this exercise, \([\cot x]\) computes the greatest integer less than or equal to \(\cot x\), ensuring integer values during the integration process.

Trigonometric Functions

Trigonometric functions are vital tools in both elementary and advanced mathematics, helping explain relationships in right-angled triangles and modeling periodic phenomena. The cotangent function, denoted as \(\cot x\), is a less commonly used trigonometric function compared to sine and cosine, but it plays a crucial role in certain calculations.

The formula for cotangent is:

• \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\)

Because \(\cot x\) is derived from sine and cosine functions, it exhibits periodic behavior repeating over intervals of \(\pi\). It has asymptotic behaviors: approaching infinity and negative infinity around \(0\) and \(\pi\), respectively. These crucial properties of \(\cot x\) affect the integral \(\int_{0}^{\pi} [\cot x] \, dx\), as seen in its shifting integer parts. Knowing the character of \(\cot x\) helps to predict where its floor -- the greatest integer -- changes, which is important for correctly partitioning the integral.