Problem 59 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int_{0}^{\pi}\left[\tan ^{-1} x\right] d x\) is equal to (where \([\cdot]\) denotes greatest integer function) (A) \(\pi\) (B) \(\tan 1\) (C) \(\pi+\tan 1\) (D) \(\pi-\tan 1\)

Step-by-Step Solution

Answer

None of these options directly match; integral evaluates to 0.

Step 1: Understanding the Problem

We're asked to evaluate the integral \( \int_{0}^{\pi}[\tan^{-1}x] \, dx \), where \([\cdot]\) is the greatest integer function, also known as the floor function. This means we need to find the area under the curve of \([\tan^{-1}x]\) from 0 to \(\pi\).

Step 2: Analyzing \(\tan^{-1}x\) Function

First, consider the behavior of \(\tan^{-1}x\) between 0 and \(\pi\). \(\tan^{-1}x\) ranges from \(\tan^{-1}(0) = 0\) to \(\tan^{-1}(\pi)\approx 1.262\). This means \(\tan^{-1}x\) is never an integer in this range.

Step 3: Applying Greatest Integer Function

Since \(\tan^{-1}x\) is always below 1 for \(0 \leq x < \pi\), the greatest integer less than or equal to \(\tan^{-1}x\) is 0 for the entire interval. Hence, \([\tan^{-1}x] = 0\) for all \(x\) in \([0, \pi)\).

Step 4: Evaluating the Integral

Given that \([\tan^{-1}x] = 0\) for the entire range of integration, the integral simplifies to \(\int_{0}^{\pi} 0 \, dx = 0.\)

Step 5: Conclusion

Since the integral evaluates to 0, it matches none of the options provided directly. However, interpreting the problem based on choices, it remains consistent that no complete unit or component falls into a whole number across the span \((0, \pi)\).

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted by \[ [x] \]is a mathematical function that rounds a real number "down" to the nearest integer less than or equal to it. Essentially, if you have a number with a decimal, the function chops off the decimal, leaving you with just the integer part. Here's what you need to know:

• If \( x = 3.7 \), then \([x] = 3\).
• If \( x = -2.3 \), then \([x] = -3\).
• It operates like the floor function in many programming languages.

This function is crucial in problems involving fractional steps, intervals, or whenever a ceiling or floor effect is desired without upward rounding. It can dramatically simplify problems by converting continuous ranges into steps.

Trigonometric Functions

Trigonometric functions are foundational to mathematics, describing relationships involving angles and lengths in right triangles, and extending to real functions defined on the unit circle. The most common ones include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). For this problem, we focus on the inverse tangent function, \( \tan^{-1}(x) \), also known as arctan. Some key points about \( \tan^{-1}(x) \):

• It gives the angle whose tangent is \( x \).
• The range is typically \( (-\pi/2, \pi/2) \).
• It is especially important in integration, due to its smooth behavior over a wide range of values.

Understanding where the function lies on the y-axis at any given \( x \)-value helps to visualize and solve integrals that involve trigonometric components. In the given problem, the function \( \tan^{-1}(x) \) never reaches an integer value throughout the interval, maintaining stability for applying the floor function.

JEE Main Mathematics

The JEE Main mathematics section is a critical standard in Indian engineering entrance examinations. It covers a broad scope of mathematical concepts, demanding proficiency in areas from algebra to calculus. Here are some tips and highlights:

• Conceptual Understanding: Mastery requires deep comprehension of basic and complex mathematical functions and their applications.
• Problem-Solving Skills: Rigorous practice is key. Understanding step-by-step methodologies is crucial, especially for integrals and function-related questions.
• Time Management: The ability to quickly identify the correct approach and execute it efficiently can significantly enhance performance.

This particular problem models typical JEE-style questions, focusing on calculus, integration, and manipulation of functions. With these foundational skills, students are better positioned to tackle a diverse array of problems, enhancing both their competency and confidence.