Animated Solution for Mathematics - Definite Integration: The value of ${\int }_{-\pi /2}^{\pi /2}\left(\frac{1}{[x]+4}\right)dx$, where $[\cdot ]$ denotes the greatest integer function, is

Welcome, future engineers! Today, we are going to peel back the layers of a seemingly intimidating integral. When you see the Greatest Integer Function (GIF) inside an integral, your first instinct might be to panic.

But let's pause. The GIF is not a monster; it is simply a staircase. It stays flat, then jumps, then stays flat again. Our goal is to walk up these stairs one step at a time.

We are tasked with evaluating the following integral:

First, let's orient ourselves. The value of $\pi$ is approximately $3.14$, so $-\frac{\pi }{2}$ is roughly $-1.57$, and $\frac{\pi }{2}$ is roughly $1.57$. The integers lying within this range are $-1,0,$ and $1$.

Because the GIF $[x]$ changes its value at every integer, we cannot treat this as one continuous function. We must break our journey into four distinct segments: $[-\frac{\pi }{2},-1)$, $[-1,0)$, $[0,1)$, and $[1,\frac{\pi }{2}]$.

Let's evaluate each segment with precision. In the first interval, $x\in [-\frac{\pi }{2},-1)$, the greatest integer is $[x]=-2$.

Our integrand becomes:

Integrating this constant over the length of the interval, which is $-1-(-\frac{\pi }{2})=\frac{\pi }{2}-1$, gives us:

Moving to the second interval, $x\in [-1,0)$, we have $[x]=-1$. The integrand is $\frac{1}{-1+4}=\frac{1}{3}$.

The length of this interval is $0-(-1)=1$. Thus:

For the third interval, $x\in [0,1)$, the greatest integer is $[x]=0$. The integrand simplifies beautifully to $\frac{1}{0+4}=\frac{1}{4}$.

The length is $1-0=1$, so:

Finally, in the last interval, $x\in [1,\frac{\pi }{2}]$, we have $[x]=1$. The integrand is $\frac{1}{1+4}=\frac{1}{5}$.

The length is $\frac{\pi }{2}-1$. So:

Now, we bring it all together. The total integral $I$ is the sum of these four parts:

To avoid the dreaded calculation error, let's group the terms. We have the $\pi$ terms:

And the constant terms:

Combining these, we get:

By finding a common denominator of $60$, we write this as $\frac{21\pi }{60}-\frac{7}{60}$, which simplifies to our final answer:

See? By breaking the problem down, we turned a complex-looking integral into a simple arithmetic exercise. Keep this mindset, and no JEE problem will ever stand in your way!