Question

$\int_{0}^{4} x[x] , dx = $
(where $[x]$ denotes the greatest integer function not greater than $x$)

• A. $17$
• B. $24$
• C. $\frac{21}{2}$
• D. $\frac{33}{2}$

Hint:

When integrating the greatest integer function, always break the integral at each integer step boundary. This turns a complex discontinuous function into simple, separate polynomial integrations that are easily evaluated.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem requires evaluating the definite integral of $x[x]$ from $0$ to $4$. The term $[x]$ represents the step-like greatest integer function, which changes its integer value abruptly at every whole number boundary.

Step 2: Key Formula or Approach:
To integrate a function containing $[x]$, we break down the net interval $[0, 4]$ into individual unit sub-intervals where the value of $[x]$ remains constant:

• For $x \in [0, 1)$, $[x] = 0$

• For $x \in [1, 2)$, $[x] = 1$

• For $x \in [2, 3)$, $[x] = 2$

• For $x \in [3, 4)$, $[x] = 3$
Using the additivity property of definite integrals: $$\int_{0}^{4} f(x)\,dx = \int_{0}^{1} f(x)\,dx + \int_{1}^{2} f(x)\,dx + \int_{2}^{3} f(x)\,dx + \int_{3}^{4} f(x)\,dx$$

Step 3: Detailed Explanation:
Substitute the piecewise values of $[x]$ into each corresponding integration segment: $$I = \int_{0}^{1} x(0) \, dx + \int_{1}^{2} x(1) \, dx + \int_{2}^{3} x(2) \, dx + \int_{3}^{4} x(3) \, dx$$ $$I = 0 + \int_{1}^{2} x \, dx + 2\int_{2}^{3} x \, dx + 3\int_{3}^{4} x \, dx$$ Now evaluate each definite integral using the standard power rule $\int x \, dx = \frac{x^2}{2}$: $$\int_{1}^{2} x \, dx = \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{4 - 1}{2} = \frac{3}{2}$$ $$2\int_{2}^{3} x \, dx = 2 \left[ \frac{x^2}{2} \right]_{2}^{3} = \left[ x^2 \right]_{2}^{3} = 9 - 4 = 5$$ $$3\int_{3}^{4} x \, dx = 3 \left[ \frac{x^2}{2} \right]_{3}^{4} = \frac{3}{2} (16 - 9) = \frac{3}{2}(7) = \frac{21}{2}$$ Sum the values of all components together: $$I = \frac{3}{2} + 5 + \frac{21}{2}$$ Combine the fractions with the common denominator 2: $$I = \frac{3 + 21}{2} + 5 = \frac{24}{2} + 5 = 12 + 5 = 17$$

Step 4: Final Answer:
The value of the definite integral is $17$, which corresponds to option (A).