Animated Solution for Mathematics - Definite Integration: Let $[t]$ denote the greatest integer $\le t$. Then the value of $8\cdot {\int }_{-1/2}^1([2x]+|x|)dx$ is .

Breaking Down the Complex

Welcome, students, to the arena of JEE Advanced. Today, we are tackling a problem that might look like a chaotic mess of symbols at first glance: $8\cdot {\int }_{-1/2}^1([2x]+|x|)dx$.

Many students see the Greatest Integer Function (GIF) and the Modulus function together and immediately feel a surge of anxiety. But I want you to take a deep breath.

In physics and mathematics, complexity is often just a collection of simple, elegant truths waiting to be separated. We are going to use the principle of linearity to turn this 'monster' into a series of simple, manageable steps.

The Linearity Strategy

Our first move is to invoke the linearity of integrals. This is our 'divide and conquer' tool. We know that the integral of a sum is the sum of the integrals.

So, we rewrite our expression as:

Let's call the first part $I_1$ and the second part $I_2$. By separating them, we stop the functions from interfering with each other. We can now analyze the 'staircase' of the GIF and the 'V-shape' of the Modulus independently.

The Staircase of $[2x]$

Focusing on $I_1={\int }_{-1/2}^1[2x]dx$, we must ask: where does this function change? The GIF $[2x]$ jumps whenever the argument $2x$ hits an integer.

In our interval $x\in [-1/2,1]$, the argument $2x$ travels from $-1$ to $2$. The integers it encounters are $0$ and $1$. This gives us critical points at $x=0$ and $x=1/2$.

We split the integral accordingly:

In the interval $[-1/2,0)$, $2x$ is in $[-1,0)$, so $[2x]=-1$. The integral becomes ${\int }_{-1/2}^0(-1)dx=-1/2$.

In $[0,1/2)$, $2x$ is in $[0,1)$, so $[2x]=0$. The integral is $0$.

Finally, in $[1/2,1)$, $2x$ is in $[1,2)$, so $[2x]=1$. The integral is ${\int }_{1/2}^1(1)dx=1/2$.

Summing these, we get $I_1=-1/2+0+1/2=0$. The staircase has perfectly balanced itself.

The V-Shape of $|x|$

Now for $I_2={\int }_{-1/2}^1|x|dx$. The modulus function is the mirror of the number line. It reflects negative values to positive ones.

We split this at the origin, $x=0$:

Evaluating the first part:

Evaluating the second part:

Adding these, $I_2=1/8+1/2=5/8$.

The Grand Finale

We have arrived at the end of our journey. The total integral $I=I_1+I_2=0+5/8=5/8$.

The problem asks for $8\cdot I$. Thus, $8\cdot (5/8)=5$.

The beauty of this result lies in the simplicity that emerges from the initial complexity. You have successfully navigated the staircase and the mirror, and the final cancellation is your reward.

Keep this mindset—break the problem down, visualize the geometry, and the math will always reveal its secrets. The final answer is 5.