Key Concepts and Formulas

1. King's Rule (Property of Definite Integrals): For a definite integral $\int_a^b f(x) dx$, the property states that $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$. This is useful for transforming the integrand into a potentially simpler form.

2. Properties of the Greatest Integer Function: For any real number $y$, $[y]$ denotes the greatest integer less than or equal to $y$. A key property is: $[y] + [-y] = \begin{cases} 0 & \text{if } y \in \mathbb{Z} \\ -1 & \text{if } y \notin \mathbb{Z} \end{cases}$

3. Integral of a Function over a Set of Measure Zero: The value of a definite integral is not affected by changing the integrand's value at a finite number of points or a countable infinity of points. These sets have "measure zero" in the context of integration.

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Step-by-Step Solution

Step 1: Define the Integral and Apply King's Rule

Let the given integral be $I$. $I = \int_0^\pi {\left[ {\cot x} \right]dx} \quad \ldots(1)$ We apply King's Rule with $a=0$ and $b=\pi$. This means we replace $x$ with $(0+\pi-x) = (\pi-x)$. $I = \int_0^\pi {\left[ {\cot (\pi - x)} \right]dx}$ Using the trigonometric identity $\cot(\pi - x) = -\cot x$, we get: $I = \int_0^\pi {\left[ { - \cot x} \right]dx} \quad \ldots(2)$

• Reasoning: King's Rule is applied to transform the integrand. The transformation of $\cot x$ to $-\cot x$ is crucial for utilizing the properties of the greatest integer function in the next step.

Step 2: Combine the Integrals

Now, we add the original integral (1) and the transformed integral (2): $I + I = \int_0^\pi {\left[ {\cot x} \right]dx} + \int_0^\pi {\left[ { - \cot x} \right]dx}$ $2I = \int_0^\pi {\left( {\left[ {\cot x} \right] + \left[ { - \cot x} \right]} \right)dx}$

• Reasoning: By adding the two expressions for $I$, we create a new integrand that is the sum of the greatest integer of a value and the greatest integer of its negative. This sum has a simplified form based on whether the value is an integer or not.

Step 3: Evaluate the Integrand Using the Greatest Integer Function Property

Consider the term inside the integral: $\left[ {\cot x} \right] + \left[ { - \cot x} \right]$. According to the property of the greatest integer function:

• If $\cot x$ is an integer, then $\left[ {\cot x} \right] + \left[ { - \cot x} \right] = 0$.
• If $\cot x$ is not an integer, then $\left[ {\cot x} \right] + \left[ { - \cot x} \right] = -1$.

In the interval $(0, \pi)$, the function $\cot x$ takes all real values from $-\infty$ to $+\infty$. This means $\cot x$ will indeed be an integer for some values of $x$ in $(0, \pi)$ (e.g., $\cot(\pi/4)=1$, $\cot(\pi/2)=0$). However, the set of points where $\cot x$ is an integer is a countable set. In the context of definite integration, changing the integrand's value at a countable number of points (a set of measure zero) does not alter the value of the integral. Therefore, for the purpose of integration over $(0, \pi)$, we can treat the integrand as: $\left[ {\cot x} \right] + \left[ { - \cot x} \right] = -1 \quad \text{for almost all } x \in (0, \pi)$

• Reasoning: This step leverages the core property of the greatest integer function. The critical insight is that the points where $\cot x$ is an integer do not contribute to the integral's value, allowing us to simplify the integrand to a constant for integration.

Step 4: Perform the Final Integration

Substitute the simplified integrand back into the expression for $2I$: $2I = \int_0^\pi {\left( { - 1} \right)dx}$ Now, we evaluate this simple integral: $2I = \left[ { - x} \right]_0^\pi$ $2I = (- \pi) - (-0)$ $2I = - \pi$ Solving for $I$: $I = - {\pi \over 2}$

• Reasoning: With the integrand simplified to a constant, the integration becomes straightforward, yielding the value of $2I$ and subsequently $I$.

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Common Mistakes & Tips

1. Misapplying the Greatest Integer Property: Ensure you correctly use the $[y] + [-y]$ property and understand its behavior for integer and non-integer values of $y$.
2. Ignoring Measure Zero: Students often get stuck trying to handle the specific points where $\cot x$ is an integer. Remember that these points have no impact on the definite integral's value.
3. Forgetting King's Rule: This rule is a powerful tool for definite integrals, especially those involving trigonometric functions. Always consider its application.

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Summary

The problem was solved by first applying King's Rule to transform the integral. This allowed us to combine the original integral with the transformed one, resulting in an integrand that was the sum of $[\cot x]$ and $[-\cot x]$. Utilizing the property of the greatest integer function, this sum was found to be $-1$ almost everywhere in the interval $(0, \pi)$. The integral then simplified to integrating $-1$ over $(0, \pi)$, which yielded $I = -\frac{\pi}{2}$.

The final answer is $\boxed{-\frac{\pi}{2}}$.