Animated Solution for Mathematics - Definite Integration: ${\int }_0^5\cos \left(\pi \left(x-\left[\frac{x}{2}\right]\right)\right)dx$, where $[t]$ denotes greatest integer less than or equal to $t$, is equal to :

The integral we are tasked to solve is:

The presence of the Greatest Integer Function, $\left[\frac{x}{2}\right]$, introduces discontinuities that require us to partition the domain $x\in [0,5]$ into manageable intervals.

The term $\left[\frac{x}{2}\right]$ remains constant until the argument $\frac{x}{2}$ reaches an integer value. Given $x\in [0,5]$, the ratio $\frac{x}{2}$ ranges from $0$ to $2.5$.

The function jumps at $\frac{x}{2}=1$ (which implies $x=2$) and $\frac{x}{2}=2$ (which implies $x=4$). Consequently, we must split the integral into three distinct zones: $[0,2)$, $[2,4)$, and $[4,5]$.

In this interval, $0\le \frac{x}{2}<1$, so $\left[\frac{x}{2}\right]=0$. The integral simplifies to:

In this interval, $1\le \frac{x}{2}<2$, so $\left[\frac{x}{2}\right]=1$. The integral becomes:

In this final interval, $2\le \frac{x}{2}<2.5$, so $\left[\frac{x}{2}\right]=2$. The integral is:

Summing the results from the three intervals, we find the total value of the integral:

The complexity of the original expression was merely a mask for a perfectly balanced system. By breaking the function at its discontinuities, we reveal that the final answer is 0.