Problem 146 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x>0\) and \(\int_{0}^{x}[x] d x=[x]\left(\frac{1}{2} A+B\right)\), where \([.]\) denotes the greatest integer function, then (A) \(A=[x]-1\) (B) \(B=x-[x]\) (C) \(A=[x]+1\) (D) \(B=x+[x]\)

Step-by-Step Solution

Answer

A = [x] + 1 and B = x - [x].

Step 1: Understand the Problem

The problem involves an integral expression and the greatest integer function, denoted by \([x]\), which represents the greatest integer less than or equal to \(x\). We are tasked with finding the values for constants \(A\) and \(B\) that satisfy the given condition.

Step 2: Evaluate the Integral

The integral given is \(\int_{0}^{x}[x] \, dx\). Since \([x]\) is a constant with respect to \(x\) on the interval where \([x]\) is constant, we can evaluate this integral as \([x](x - 0) = [x]x\).

Step 3: Set Up the Equation

The equation given in the problem is \([x]x = [x]\left(\frac{1}{2} A + B\right)\). This indicates a relationship involving \([x]\) as a common factor on both sides.

Step 4: Simplify and Compare Coefficients

Divide both sides of the equation \([x]x = [x]\left(\frac{1}{2} A + B\right)\) by \([x]\) assuming \([x]eq 0\), resulting in \(x = \frac{1}{2}A + B\). Since both sides should match, this equation will help us determine \(A\) and \(B\).

Step 5: Determine A and B

From \(x = \frac{1}{2}A + B\), and by comparing options for \(A\) and \(B\), note the choices: (A) and (C) suggest \(A\) as \([x] \pm 1\), and (B) and (D) suggest \(B\) as combinations of \(x\) and \([x]\). The expression \( \frac{1}{2}([x] + 1) + (x - [x]) = \frac{1}{2}[x] + \frac{1}{2} + x - [x]\) simplifies to \(x\), indicating that \(A = [x] + 1\) and \(B = x - [x]\).

Step 6: Verify the Solution

Substitute \(A = [x] + 1\) and \(B = x - [x]\) back into \([x](\frac{1}{2}A + B) = [x]x\):\([x](\frac{1}{2}([x]+1) + (x-[x])) = [x](\frac{1}{2}[x] + \frac{1}{2} + x - [x]) = [x]x\).The equation holds, confirming the values of \(A\) and \(B\) are correct.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, often denoted by \([x]\), gives the largest integer less than or equal to a given number \x\. This is also known as the floor function. It plays a unique role in various mathematical applications by simplifying expressions, especially when dealing with integrals or functions where piecewise definitions are involved.

• For example, if \x = 2.7\, then \([x] = 2\), because 2 is the largest integer less than 2.7.
• Similarly, if \x = -1.4\, then \([x] = -2\), because -2 is the largest integer less than -1.4.

Using this function is crucial in problems involving definite integrals where the function value may not maintain constancy across an interval. It's also useful in programming and algorithm designs where integer partitioning is required. The key to mastering the Greatest Integer Function is practice, as it can initially be counterintuitive due to its behavior with negative numbers.

Integral Calculus

Integral Calculus is one of the two main branches of calculus. It focuses on finding the total size or value, such as areas under curves or the accumulation of quantities. Integrals can be thought of as the "sum over a continuum," providing a way to calculate quantities that accumulate continuously rather than discretely.

• The symbol for integration is \(\int\), and it requires an integrand or function, along with the variable of integration.
• Definite integrals have limits of integration, which define the interval over which the integration is performed. They yield a numerical value representing the area under a curve between those limits.
• Indefinite integrals, on the other hand, do not have limits and usually require an additional constant of integration.

Integral calculus is not only about calculating areas but also includes applications in physics, engineering, and beyond. Understanding its core principles, like the Fundamental Theorem of Calculus, is essential for solving real-world problems involving rates of change and accumulation.

Definite Integrals

Definite Integrals are a type of integral calculus where the limits of integration are specified. They compute the accumulation of quantities that can be represented geometrically as the area under a curve within certain bounds. For instance, if we're looking at an integral \(\int_{a}^{b} f(x) \, dx\), \a\ and \b\ are the limits of integration.

• The significance of definite integrals in mathematical analysis is crucial as they provide exact values rather than expressions with constants of integration.
• By evaluating the area under the curve \f(x)\ between \a\ and \b\, definite integrals can solve numerous practical problems ranging from physics to statistics.
• They are often solved using the properties of antiderivatives and applying the Fundamental Theorem of Calculus, which relates differentiation and integration.

An important note is when functions like the Greatest Integer Function are involved, care must be taken since the function value is piecewise constant over intervals. This changes the way we approach the integration, often simplifying the task by breaking it into manageable pieces for each constant segment.