Animated Solution for Mathematics - Definite Integration: The integral ${\int }_{-1/2}^{1/2}([x]+\ln (\frac{1+x}{1-x}))dx$ equal to

Welcome, fellow traveler on the JEE Advanced path. Today, we are going to dissect an integral that looks intimidating at first glance but hides a beautiful, elegant secret.

We are tasked with evaluating the integral:

At first, you might feel the urge to dive into complex substitutions or integration by parts, but pause. In the world of JEE, the most powerful tool is often observation. Let us break this problem down into its two distinct personalities: the Greatest Integer Function (GIF) and the logarithmic function.

The Logarithmic Trap

We begin by using the linearity property of integrals, splitting our expression into two manageable pieces:

Let us focus on the second part, $h(x)=\ln \left(\frac{1+x}{1-x}\right)$. Whenever you see an integral with symmetric limits, like $[-a,a]$, your mathematical intuition should immediately scream: "Check for symmetry!"

Is the function odd or even? Let us test it by substituting $-x$ for $x$:

Now, recall the logarithmic identity $\ln (a/b)=-\ln (b/a)$. Applying this, we get $h(-x)=-\ln \left(\frac{1+x}{1-x}\right)=-h(x)$.

This confirms that $h(x)$ is an odd function. Because the integral of an odd function over a symmetric interval is always zero, the entire logarithmic term vanishes into thin air.

The GIF Staircase

Now, we are left with the first part:

The Greatest Integer Function is not a smooth curve; it is a staircase. It remains constant between integers and jumps at every integer value. In our interval $[-1/2,1/2]$, the only point of discontinuity is $x=0$.

To handle this, we must split the integral at the point of the jump:

For the first sub-interval, $x\in [-1/2,0)$, the value of $[x]$ is $-1$. For the second sub-interval, $x\in [0,1/2]$, the value of $[x]$ is $0$. This simplifies our integral significantly:

The Final Synthesis

Evaluating these is straightforward. The first integral is:

The second integral is simply ${\int }_0^{1/2}0dx=0$. Adding these together, we get our final result:

We have navigated the trap of the log function and the discontinuity of the GIF, arriving at our answer with logic and precision. Remember, in JEE Advanced, it is not just about calculating; it is about seeing the structure of the problem before you even pick up your pen.

The final answer is $-1/2$.