Animated Solution for Mathematics - Definite Integration: If $[\cdot ]$ represents the greatest integer function, then the value of $\left|{\int }_0^{\sqrt{\pi /2}}[[x^2]-\cos x]dx\right|$ is ____

The problem asks us to evaluate the expression:

At first glance, this looks like a complex expression involving nested brackets. However, we can simplify this by applying fundamental properties of the greatest integer function.

The key to this problem lies in the property $[n+f(x)]=n+[f(x)]$, where $n$ is an integer. Here, $[x^2]$ acts as our integer $n$.

Because $x^2$ is a real number, $[x^2]$ is always an integer. This allows us to rewrite the integrand as:

Suddenly, the expression is tamed. We have successfully separated the integer part from the transcendental part.

We must now examine the interval $[0,\sqrt{\frac{\pi }{2}}]$. Since $\sqrt{\frac{\pi }{2}}\approx 1.25$, we analyze the behavior of $[x^2]$ and $[-\cos x]$ within this range.

For $[x^2]$, the value changes whenever $x^2$ reaches an integer. In our interval, $x^2$ ranges from $0$ to $\frac{\pi }{2}\approx 1.57$. The only integer it crosses is $1$, which occurs at $x=1$. This is our critical split point.

For $[-\cos x]$, we observe that in the interval $[0,\sqrt{\frac{\pi }{2}}]$, $\cos x$ is always between $0$ and $1$. Therefore, $-\cos x$ lies in the interval $[-1,0)$. The greatest integer of any value in $[-1,0)$ is always $-1$. This term is a constant.

We split the integral at the critical point $x=1$:

In the first interval $[0,1]$, we have $[x^2]=0$ and $[-\cos x]=-1$. The integrand simplifies to $-1$:

In the second interval $[1,\sqrt{\frac{\pi }{2}}]$, we have $[x^2]=1$ and $[-\cos x]=-1$. The integrand simplifies to $1-1=0$:

Adding these results together, we obtain $-1+0=-1$. Finally, taking the absolute value as required by the problem: