Key Concepts and Formulas

1. Property of Definite Integrals with Symmetric Limits: For an integrable function $f(x)$, $\int_{-a}^a f(x) \, dx = \int_0^a [f(x) + f(-x)] \, dx$ This property is extremely useful for integrals with symmetric limits, often simplifying the integrand.

2. Property of the Greatest Integer Function (GIF): For any real number $y$, $[y] + [-y] = \begin{cases} 0 & \text{if } y \in \mathbb{Z} \\ -1 & \text{if } y \notin \mathbb{Z} \end{cases}$ The behavior of $[y] + [-y]$ depends crucially on whether $y$ is an integer or not.

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Step-by-Step Solution

Let the given integral be $I$. $I = \int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$

Step 1: Apply the Property of Symmetric Limits

• Why this step? The integration limits are from $-2$ to $2$, which are symmetric about $0$. This suggests using the property $\int_{-a}^a f(x) \, dx = \int_0^a [f(x) + f(-x)] \, dx$ to simplify the integral.

Let $f(x) = \frac{\sin^2 x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}}$. We need to find $f(-x)$. $f(-x) = \frac{\sin^2 (-x)}{\left[ \frac{-x}{\pi} \right] + \frac{1}{2}}$ Since $\sin(-x) = -\sin x$, $\sin^2(-x) = (-\sin x)^2 = \sin^2 x$. So, $f(-x) = \frac{\sin^2 x}{\left[ -\frac{x}{\pi} \right] + \frac{1}{2}}$ Applying the property, the integral becomes: $I = \int_0^2 \left( f(x) + f(-x) \right) \, dx$ $I = \int_0^2 \left( \frac{\sin^2 x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}} + \frac{\sin^2 x}{\left[ -\frac{x}{\pi} \right] + \frac{1}{2}} \right) \, dx$

Step 2: Analyze the Greatest Integer Function Term

• Why this step? The denominators involve the greatest integer function. To simplify the sum of the two fractions, we need to understand the relationship between $\left[ \frac{x}{\pi} \right]$ and $\left[ -\frac{x}{\pi} \right]$ within the integration interval $[0, 2]$.

Consider the term $y = \frac{x}{\pi}$. For $x \in (0, 2]$, the range of $y$ is $\left( 0, \frac{2}{\pi} \right]$. Since $\pi \approx 3.14159$, we have $\frac{2}{\pi} \approx 0.6366$. Thus, for $x \in (0, 2]$, $\frac{x}{\pi}$ lies in the interval $(0, 0.6366]$. In this interval, $\frac{x}{\pi}$ is never an integer. Therefore, we can use the GIF property: $\left[ \frac{x}{\pi} \right] + \left[ -\frac{x}{\pi} \right] = -1$. From this, we get $\left[ -\frac{x}{\pi} \right] = -1 - \left[ \frac{x}{\pi} \right]$.

Step 3: Substitute and Simplify the Integrand

• Why this step? We substitute the expression for $\left[ -\frac{x}{\pi} \right]$ into the integral to see if the integrand simplifies.

Substitute $\left[ -\frac{x}{\pi} \right] = -1 - \left[ \frac{x}{\pi} \right]$ into the second term of the integrand: $\frac{\sin^2 x}{\left[ -\frac{x}{\pi} \right] + \frac{1}{2}} = \frac{\sin^2 x}{\left( -1 - \left[ \frac{x}{\pi} \right] \right) + \frac{1}{2}}$ Simplify the denominator: $-1 - \left[ \frac{x}{\pi} \right] + \frac{1}{2} = -\frac{1}{2} - \left[ \frac{x}{\pi} \right] = - \left( \left[ \frac{x}{\pi} \right] + \frac{1}{2} \right)$ Now, substitute this back into the integral expression: $I = \int_0^2 \left( \frac{\sin^2 x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}} + \frac{\sin^2 x}{- \left( \left[ \frac{x}{\pi} \right] + \frac{1}{2} \right)} \right) \, dx$ $I = \int_0^2 \left( \frac{\sin^2 x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}} - \frac{\sin^2 x}{\left[ \frac{x}{\pi} \right] + \frac{1}{2}} \right) \, dx$ The two terms in the integrand cancel each other out: $I = \int_0^2 (0) \, dx$

Step 4: Evaluate the Integral

• Why this step? The integral of zero over any finite interval is zero.

$I = 0$

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Common Mistakes & Tips

• GIF Condition Check: Always verify if the argument of the GIF is an integer or not within the integration limits. Incorrectly applying $[y] + [-y] = -1$ when $y$ is an integer (or vice-versa) is a common error.
• Symmetric Limit Property: Recognizing and applying the $\int_{-a}^a f(x) \, dx = \int_0^a [f(x) + f(-x)] \, dx$ property is crucial for simplifying integrals with symmetric bounds.
• Algebraic Simplification: Be meticulous with algebraic manipulations, especially when dealing with denominators that involve GIF terms, to avoid sign errors.

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Summary

The integral was evaluated by first applying the property of definite integrals with symmetric limits, transforming the integral from $[-2, 2]$ to $[0, 2]$ by considering $f(x) + f(-x)$. The key step involved analyzing the greatest integer function $\left[ \frac{x}{\pi} \right]$. For $x \in (0, 2]$, $\frac{x}{\pi}$ is never an integer, allowing us to use the property $\left[ y \right] + \left[ -y \right] = -1$. This led to the simplification of the integrand to zero, making the value of the integral zero.

The final answer is $\boxed{0}$.