How to Integrate {x}^[x] from Limits 0 to 3

Integrating the function {x}^{[x]} from 0 to 3, where {x} is the fractional part function and [x] is the greatest integer function, involves breaking the problem into several steps. This process not only provides a clear understanding of the behavior of the function but also simplifies the integration process. In this article, we will walk through each step in detail.

Understanding the Functions

The fractional part function {x} is defined as (x - [x]), where [x] is the greatest integer function, which returns the largest integer less than or equal to x. This means for any real number x, {x} is the part of x that remains after subtracting the greatest integer that is less than or equal to x.

Interval Breakdown

To evaluate the integral, we break the interval from 0 to 3 into subintervals where [x] is constant. This simplifies the integration by allowing us to work with simpler functions in each segment.

Step 1: Interval from 0 to 1

In the interval 0 ≤ x

Step 2: Interval from 1 to 2

In the interval 1 ≤ x

Step 3: Interval from 2 to 3

In the interval 2 ≤ x ≤ 3, [x] 2. Therefore, the function simplifies to {x}^{[x]} {x}^2 x - 2^2 x - 4. The integral over this interval also follows a similar pattern.

Setting Up the Integral

Using the interval breakdown, we can express the integral as the sum of integrals over these segments.

Mathematically, this is:

( int_{0}^{3} {x}^{[x]} , dx int_{0}^{1} {x}^{0} , dx int_{1}^{2} {x - 1} , dx int_{2}^{3} {x - 4} , dx )

Calculation of Each Integral

Let's compute each of the integrals step by step.

Integral from 0 to 1

( int_{0}^{1} 1 , dx [x]_{0}^{1} 1 - 0 1 )

Integral from 1 to 2

( int_{1}^{2} (x - 1) , dx left[ frac{(x - 1)^2}{2} right]_{1}^{2} left[ frac{(2 - 1)^2}{2} right] - left[ frac{(1 - 1)^2}{2} right] frac{1}{2} - 0 frac{1}{2} )

Integral from 2 to 3

( int_{2}^{3} (x - 4) , dx left[ frac{(x - 4)^2}{3} right]_{2}^{3} left[ frac{(3 - 4)^2}{3} right] - left[ frac{(2 - 4)^2}{3} right] frac{1}{3} - 0 frac{1}{3} )

Summing the Results

Now, we add all the results together:

( 1 frac{1}{2} frac{1}{3} )

To add these fractions, we find a common denominator, which is 6:

( 1 frac{6}{6}, frac{1}{2} frac{3}{6}, frac{1}{3} frac{2}{6} )

Adding them together:

( frac{6}{6} frac{3}{6} frac{2}{6} frac{11}{6} )

Final Result

Thus, the value of the integral is:

( int_{0}^{3} {x}^{[x]} , dx frac{11}{6} )

Graphical Interpretation

It is often easier to understand the integral by drawing the graph of the function. The graph consists of shifted versions of the functions y 1, y x, and y x^2. By recognizing this, we can simplify the integration process, making the problem much more straightforward.

The integral can be evaluated as:

( int_{0}^{3} 1 , dx int_{0}^{1} x , dx int_{0}^{1} x^2 , dx )

This simplifies to:

( 1 frac{1}{2} frac{1}{3} frac{11}{6} )