Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the integral \(\int_{1}^{e}\left[\frac{\log x}{3}\right] d x\), where \([\cdot]\) denotes the greatest integer function, is (A) 0 (B) \(e^{6}-e^{3}\) (C) \(e^{6}+e^{3}\) (D) \(e^{3}-e^{6}\)

Step-by-Step Solution

Answer

The integral evaluates to 0, which is option (A).

Step 1: Understanding the Greatest Integer Function

The greatest integer function, represented as \( [\log x/3] \), returns the largest integer less than or equal to the value of \( \log x/3 \). This means we need to find intervals where \( \log x/3 \) is an integer value or crosses an integer boundary.

Step 2: Determine the Integer Ranges

We evaluate \( \log x = \log 1 = 0 \) at \( x = 1 \) and \( \log x = 1 \) at \( x = e \). Therefore, at \( x = 1 \), \( \log x/3 = 0 \), and \( x = e \), \( \log x/3 = 1/3 \). In the interval \( [1, e] \), \( [\log x/3] = 0 \) since the value 1/3 for \( x = e \) is still rounded down to 0.

Step 3: Evaluate the Integral

Since \( [\log x/3] = 0 \) over the entire interval from 1 to \( e \), the integral simplifies to \( \int_{1}^{e} 0 \, dx = 0 \).

Step 4: Select the Correct Option

Based on our evaluation in previous steps, the integral evaluates to 0, which corresponds to option (A).

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted by the symbol \(\lfloor \cdot \rfloor\), returns the largest integer that is less than or equal to a given number. This means, for any real number \(x\), \(\lfloor x \rfloor\) is the greatest integer which is either equal to or less than \(x\). For example:

• \(\lfloor 2.7 \rfloor = 2\)
• \(\lfloor -1.3 \rfloor = -2\)

This concept is crucial in calculus problems where functions need to be approximated or simplified.
When dealing with functions like \([\frac{\log x}{3}]\), the aim is to find integer values within specific ranges.
In the provided exercise, knowing that \(\log x\) changes continuously with \(x\), the intervals where \([\frac{\log x}{3}]\) changes integer values need to be identified. The rounding down at integer boundaries provides insight into how the function behaves across an interval, simplifying the integration process.

Logarithmic Functions

Logarithmic functions involve the logarithm, which is the inverse of exponentiation. The base of the logarithm is pivotal in determining the nature of the function. The common logarithm, \(\log\), is typically base 10, but in calculus, the natural logarithm, denoted as \(\ln x\), or \(\log x\) with base \(e\), is most common.
The fundamental property of logarithms that assists in solving calculus problems is that \(\log (a \cdot b) = \log a + \log b\). This property highlights the logarithm's ability to transform multiplication into addition, simplifying the process of integration and differentiation.
In the exercise, \(\log x\) with natural base \(e\) is employed. Here, \(x = 1\) results in \(\log 1 = 0\), and \(x = e\) results in \(\log e = 1\). Understanding these basic transformations is key in analyzing how the function behaves across its domain and lays the groundwork for applying other calculus concepts effectively.

Calculus Problems

Problems in calculus often require a blend of techniques from both integration and differentiation. Here, we focus on the integration aspect, particularly definite integrals. A definite integral \(\int_{a}^{b} f(x) \, dx\) computes the area under the curve of \(f(x)\), from \(x = a\) to \(x = b\).
In this exercise, the function is \(\left[\frac{\log x}{3}\right]\), which incorporates the greatest integer function. The challenge is to accurately account for the intervals wherein this function maintains constant integer values across the integration domain.
To evaluate such integrals, identify intervals over which the function within the brackets doesn't change its integer value, then integrate this constant value over the interval's length. Solving these problems typically involves first simplifying the function within the integral to better understand its behavior over the specified domain.