Animated Solution for Mathematics - Definite Integration: The value of ${\int }_{-\pi /2}^{\pi /2}\frac{dx}{[x]+[\sin x]+4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is :

We are tasked with evaluating the integral:

The presence of the Greatest Integer Function (GIF) symbols $[x]$ and $[\sin x]$ indicates that the integrand is a piecewise constant function. To solve this, we must identify the points of discontinuity within the domain $[-\pi /2,\pi /2]\approx [-1.57,1.57]$.

The function $[x]$ jumps at integers $-1,0,$ and $1$. The function $[\sin x]$ jumps when $\sin x$ equals an integer, which occurs at $x=-\pi /2,0,$ and $\pi /2$. Combining these, we partition the domain into four distinct intervals: $[-\pi /2,-1)$, $[-1,0)$, $[0,1)$, and $[1,\pi /2)$.

We evaluate the integral across each identified interval:

1. Interval $[-\pi /2,-1)$: Here, $[x]=-2$ and $[\sin x]=-1$. The denominator is $-2-1+4=1$.

2. Interval $[-1,0)$: Here, $[x]=-1$ and $[\sin x]=-1$. The denominator is $-1-1+4=2$.

3. Interval $[0,1)$: Here, $[x]=0$ and $[\sin x]=0$. The denominator is $0+0+4=4$.

4. Interval $[1,\pi /2)$: Here, $[x]=1$ and $[\sin x]=0$. The denominator is $1+0+4=5$.

To find the total value of the integral, we sum the results from each interval:

Grouping the terms involving $\pi$ and the constant terms:

Combining these, we obtain the final result: