Problem 118 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int_{1}^{4}(\\{x\\})^{[x]} d x\), where \(\\{\cdot\\}\) and \([\cdot]\) denote the fractional part and greatest integer function, respectively, is equal to (A) 1 (B) \(\frac{12}{13}\) (C) \(\frac{13}{12}\) (D) \(\frac{6}{7}\)

Step-by-Step Solution

Answer

The answer is (C) \(\frac{13}{12}\).

Step 1: Analyze the Expression

The problem involves an integral from 1 to 4 of the expression \((\{x\})^{[x]}\). Here, \(\{x\}\) is the fractional part of \(x\), which means \(x\) minus the greatest integer less than or equal to \(x\). The \([x]\) is the greatest integer function (floor function), giving the largest integer less than or equal to \(x\).

Step 2: Evaluate the Integral by Splitting Intervals

To solve \(\int_{1}^{4}(\{x\})^{[x]} dx\), first recognize that \([x]\) will be constant over each integer interval. Thus, split the integral: \(\int_{1}^{2}(\{x\})^{1} dx + \int_{2}^{3}(\{x\})^{2} dx + \int_{3}^{4}(\{x\})^{3} dx.\)

Step 3: Compute Each Integral Separately

Compute each integral separately: \(1 \leq x < 2\) gives \(\int_{1}^{2}x-1 dx\) which evaluates to \([\frac{x^2}{2} - x]_{1}^{2} = \frac{1}{2}.\)\(\) For \(2 \leq x < 3\), the integral is \(\int_{2}^{3}(x-2)^2 dx\) which evaluates to \([\frac{x^3}{3} - 2x^2 + 4x]_{2}^{3} = \frac{1}{3}.\)\(\) For \(3 \leq x < 4\), the integral is \(\int_{3}^{4}(x-3)^3 dx\) which evaluates to \([\frac{x^4}{4} - 3x^3 + 9x^2 - 27x]_{3}^{4} = \frac{1}{4}.\)\(\).

Step 4: Sum the Results

Add the results of the integrals from each interval to find the total: \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4}.\)

Step 5: Simplify the Total

Convert \(\frac{1}{2}, \frac{1}{3},\) and \(\frac{1}{4}\) to have a common denominator, which is 12, and sum: \(\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}.\)

Key Concepts

Fractional Part Function

The fractional part function is an interesting component often encountered in integral calculus. This function, denoted by \( \{x\} \), refers to the "leftover" part of a real number after removing its integer part. Essentially, it represents the difference between the number and its greatest integer less than or equal to it.
For instance, if you take \( x = 3.75 \), the fractional part \( \{3.75\} \) is 0.75, as the greatest integer less than or equal to 3.75 is 3.
This function helps in situations where you want to focus on the decimal part independent from the integer part.

• This function is periodic with a period of 1.
• It outputs a value between 0 and less than 1.

The fractional part function provides unique challenges in integration, often requiring the problem to be broken down into intervals where this behavior remains consistent.
In our context, since the fractional part changes at every integer, we split the integral at integers.

Greatest Integer Function

The greatest integer function, also known as the floor function and denoted by \( [x] \), rounds down a real number to the nearest integer less than or equal to it. This serves as the foundation for understanding intervals in calculus.
It is useful when dealing with step functions or piecewise constants, as it provides a simplistic representation of numbers splitting them into integer portions, making it essential for discretized data in integrals.

• For example, for any real number \( x \), \( [2.7] = 2 \) and \( [-3.2] = -4 \).
• It is a piecewise constant function, meaning it remains constant through each interval but changes at every integer.

In our exercise, the greatest integer function controls how the expression \( (\{x\})^{[x]} \) behaves, remaining constant over ranges like [1,2), [2,3), and [3,4).
Each interval has a unique power level dictated by \([x]\), which simplifies integral calculations.

Integration Techniques

Integration techniques are essential tools in calculus, allowing us to find areas under curves and solve complex problems involving continuous functions.
One useful approach is breaking down complex integrals into simpler parts, especially when dealing with functions like the fractional part and the greatest integer function.
Splitting the integral into separate regions, often defined by changes in the greatest integer function, allows easier computation.

• In our exercise, each integral is calculated separately over intervals where the expression \( (\{x\})^{[x]} \) remains well-defined and consistent.
• This method leverages the property that these functions are easier to manage piecewise.

As demonstrated in the step-by-step solution, integrating each segment individually and then summing the results simplifies computations.
For instance, the sum of integrals over [1,2), [2,3), and [3,4) results in the final value by leveraging the powers defined by \([x]\) and the smooth continuity within each range.
This techinque exemplifies how integration can be made manageable with strategic planning and understanding of function behavior.