Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \([x]\) denotes the greatest integer \(\leq x\), then the value of \(\int_{4}^{10} \frac{\left[x^{2}\right]}{\left[x^{2}-28 x+196\right]+\left[x^{2}\right]} d x\) is (A) 3 (B) 2 (C) 1 (D) 0

Step-by-Step Solution

Answer

The answer is not available because the integral calculations are inaccurate with current information. Please revise the limits or the setup.

Step 1: Simplify the Denominator

Notice that the denominator is \([x^2 - 28x + 196] + [x^2]\). Simplifying \[x^2 - 28x + 196\] gives \((x-14)^2\). Since this is always a non-negative expression, the floor value is simply \[0\] when \((x - 14)^2 \leq 0\), but here since \[x\] is between 4 and 10, \[x - 14\] is always negative. Thus, the whole expression for each integer is \[x^2 - [x^2] + 0 = 0\]. This simplifies to just \[x^2\] as the denominator.

Step 2: Define the Function

The function inside the integral is \ \( f(x) = \frac{[x^2]}{[x^2] + 0} = 1\) for integer values of \[x\]. The reason being that both the numerator and denominator are equal.

Step 3: Evaluate the Integral from 4 to 10

Since \(f(x) = 1\) for all integer \[x\] from 4 to 9 (as \[10\] is not included in the range as an integer) because the greatest integer function remains constant at integer points, we simply count the number of these integer points. Thus, the value of \(\int_4^{10} 1 \, dx\) equals \(6\).

Key Concepts

Greatest Integer Function

The greatest integer function is often denoted by \([x]\). It takes any real number \((x)\) as input and gives the greatest integer less than or equal to \((x)\) as output. For example, if \(x = 2.7\), then \([2.7] = 2\). The function is also referred to as the floor function because, conceptually, it "floors" \((x)\) to the nearest integer less than or equal to it.
The greatest integer function is a step function, which means it jumps at every integer. Each step occurs exactly at integer values, representing a discontinuous point where the function value suddenly shifts to the next integer. This is critical in analysis because it changes the behavior of functions that interact with it, especially when dealing with definite integrals over intervals.

Floor Function

The floor function, often symbolized as \(\lfloor x \rfloor\), essentially operates as described by the greatest integer function. Both represent ways to handle fractional parts of numbers by rounding them down to the nearest whole number. This function is identifiable by a characteristic drop in value at each transition between integer values.
Often used in mathematical analysis, the floor function provides a way to control or partition real numbers, simplifying complex algebraic and calculus problems. In definite integrals, expressions involving the floor function can often be simplified over integer-sized subintervals.

Integration by Simplification

Integration by simplification involves altering and reducing complex integral expressions to forms that are easier to integrate. In the case of the given exercise, this means considering how expressions involving the floor function can be treated over specified intervals.
By substituting and simplifying the expression within the integral, especially the denominator, the complex integral is often transformed into a much simpler problem. This eliminates complicated functions or allows replacing them with constants. The solver can then directly evaluate simpler functions over the desired interval. This strategy significantly reduces computational workload, particularly when entire sections of an integral evaluate uniformly, as in the given problem.

Interval Analysis

Interval analysis in integration assesses how a function behaves over specific ranges of its variables. For this exercise, it involves understanding how the greatest integer function or floor function operates across intervals of definite integrals, where evaluation of the function simplifies step-by-step over those intervals.
When analyzing an interval from 4 to 10, the function within the integral simplifies to constant values on integer subdomains due to these floor functions. Such intervals allow for specific integer evaluations simplifying exact integral computation, as the function behavior is known over each small partition. Correctly interpreting these intervals and the way floor functions modify them is key to solving such integrals effectively.