JEE Mathematics 2023 Question with Solution

Given: $[t]$ denotes the greatest integer function, and we need to evaluate

Find: The numerical value of the integral.

From the solution, first split the integral into two parts:

Step 1: Evaluate the part involving $[\csc x]$. On the interval $\left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$, the solution states:

• from $\frac{\pi}{6}$ to $\frac{\pi}{2}$, $[\csc x] = 2$,
• from $\frac{\pi}{2}$ to $\frac{5\pi}{6}$, $[\csc x] = 1$.

Hence,

Evaluating,

Step 2: Evaluate the part involving $[\cot x]$. The solution uses the substitution $x \to \pi - x$ and writes:

If this integral is denoted by $I$, then the solution writes:

So it simplifies to

Then the solution concludes:

Step 3: Combine the results exactly as shown in the solution:

Therefore, the value of the integral is $14$.

Given: The integral contains greatest integer functions of $\csc x$ and $\cot x$ over $\left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$. Find: The required numerical value.

The key idea stated in the hint is to split the interval into subintervals where the floor values stay constant. In the provided solution, the part involving $[\csc x]$ is evaluated by observing constant values on subintervals, while the part involving $[\cot x]$ is handled by symmetry through the substitution $x \to \pi - x$.

Using those exact observations from the solution leads to the final reported value $14$. Hence the required numerical answer is $14$.