Animated Solution for Mathematics - Definite Integration: ${\int }_0^{\pi }[\cot x]dx$, where $[.]$ denotes the greatest integer function, is equal to:

Imagine you are standing before a complex, jagged landscape. You are asked to calculate the area under a curve that looks like a staircase, constantly jumping from one integer value to another.

This is the challenge of evaluating ${\int }_0^{\pi }[\cot x]dx$. At first glance, it looks like a nightmare of infinite steps.

But in the world of JEE Advanced mathematics, whenever you see a problem that looks impossibly tedious, there is almost always a hidden symmetry waiting to be discovered. Let us embark on this journey together.

If you try to evaluate this integral by breaking it into intervals where $[\cot x]$ is constant, you will quickly find yourself lost. The function $\cot x$ ranges from positive infinity near $x=0$ to negative infinity near $x=\pi$.

It hits every integer value along the way. Trying to sum the areas of these infinite, tiny rectangles is a path to frustration.

We need a more elegant approach. We need to look at the problem from a different perspective.

A Mathematical Lifesaver

Whenever you encounter a definite integral with limits from $0$ to $a$, your intuition should immediately scream, "King's Property!" This property is one of the most powerful tools in your arsenal:

It allows us to transform the variable $x$ into $a-x$ without changing the value of the integral. Let us apply this to our integral $I={\int }_0^{\pi }[\cot x]dx$.

By replacing $x$ with $\pi -x$, we get $I={\int }_0^{\pi }[\cot (\pi -x)]dx$. Now, recall the trigonometric identity $\cot (\pi -x)=-\cot x$.

Suddenly, our integral transforms into $I={\int }_0^{\pi }[-\cot x]dx$. This is the breakthrough we needed!

Now, we have two expressions for the same integral $I$. Let us add them together:

Look closely at the integrand. We have the sum of the greatest integer of a number and the greatest integer of its negative.

There is a beautiful property of the Greatest Integer Function: $[t]+[-t]=-1$ for any $t$ that is not an integer. Since $\cot x$ is an integer only at discrete, isolated points, these points do not affect the value of the definite integral.

Therefore, for almost all $x$ in the interval $(0,\pi )$, the expression $[\cot x]+[-\cot x]$ is simply $-1$. Our terrifying step function has vanished, replaced by a simple constant!

We are left with the simple integral:

This is a basic calculation:

Finally, dividing by $2$, we arrive at our answer:

This problem is a perfect reminder that in mathematics, brute force is rarely the answer. By embracing symmetry and using the right properties, we can turn a seemingly impossible problem into a moment of pure, elegant clarity.