Animated Solution for Mathematics - Definite Integration: The value of the integral ${\int }_{-2}^2\frac{{\sin }^{2}x}{[x/\pi ]+1/2}dx$ (where $[x]$ denotes the greatest integer less than or equal to $x$) is :

Conquering the Symmetric Integral

Welcome, future engineers! Today, we are going to dissect a problem that perfectly encapsulates the philosophy of the JEE Advanced examination.

Many students, upon seeing an integral like , immediately reach for their pens, ready to break the interval into pieces and perform a grueling calculation.

But stop! Before you dive into the algebra, take a breath. Look at the limits and the structure. In the world of competitive mathematics, the most powerful tool you possess is not your speed of calculation, but your ability to observe.

The Power of Symmetry

Whenever you encounter an integral with symmetric limits, specifically from $-a$ to $a$, your mathematical radar should immediately ping. This is due to the beautiful property of odd functions.

If a function $f(x)$ is odd—meaning $f(-x)=-f(x)$—then the integral is identically zero.

Think about the geometry: an odd function has rotational symmetry about the origin. The area accumulated on the negative side of the $x$-axis is exactly equal in magnitude but opposite in sign to the area on the positive side, causing them to cancel out perfectly.

The Anatomy of the Integrand

Let us break this function down into its two components: the numerator and the denominator.

First, the numerator: ${\sin }^{2}x$. We know that $\sin (-x)=-\sin x$. When we square this, we get $(-\sin x{)}^2={\sin }^{2}x$.

The numerator is an even function; it remains unchanged when we replace $x$ with $-x$. This means the "oddness" of the entire function depends entirely on the denominator.

Now, consider the denominator: $D(x)=[x/\pi ]+1/2$. This is where the Greatest Integer Function (GIF) resides. To test for oddness, we must evaluate $D(-x)=[-x/\pi ]+1/2$.

The GIF Transformation

Here is where we use our secret weapon: the property of the Greatest Integer Function. For any non-integer $y$, we have the identity $[-y]=-[y]-1$.

In our case, $y=x/\pi$. Since our interval is $[-2,2]$, the value of $x/\pi$ ranges from approximately $-0.63$ to $0.63$. Within this range, $x/\pi$ is only an integer at $x=0$, so the property holds for all other values.

Let us apply it to the denominator:

Using our property, we substitute $[-x/\pi ]$ with $-[x/\pi ]-1$:

Simplifying the constants, $-1+1/2$ results in $-1/2$:

Now, factor out the negative sign:

The denominator has transformed into the negative of its original self.

We have our numerator, which stayed the same, and our denominator, which flipped its sign. When we combine them:

We have mathematically proven that $f(x)$ is an odd function. Because the limits of integration are symmetric from $-2$ to $2$, the positive area and the negative area cancel out perfectly.

Therefore, the final result is:

There you have it. No complex integration, no piecewise summation, just pure, elegant logic. This is the beauty of mathematics—when you understand the properties of the functions you are working with, the problem solves itself.