Animated Solution for Mathematics - Definite Integration: ${\int }_0^2\left(|2x^2-3x|+[x-\frac{1}{2}]\right)dx$, where $[t]$ is the greatest integer function, is equal to:

Our integral is defined as:

The first principle of advanced calculus is 'divide and conquer.' Because the integral is a linear operator, we can split this into two distinct problems: $I=I_1+I_2$.

Here, $I_1={\int }_0^2|2x^2-3x|dx$ and $I_2={\int }_0^2[x-\frac{1}{2}]dx$. By separating them, we stop the functions from interfering with each other.

Let us look at $I_1$. The expression inside the modulus is a quadratic: $f(x)=2x^2-3x$. If we factor this, we get $x(2x-3)$, which tells us the roots are at $x=0$ and $x=\frac{3}{2}$.

This is an upward-opening parabola. Between $0$ and $\frac{3}{2}$, the parabola dips below the x-axis, meaning the value is negative. The modulus function, being the guardian of positivity, will flip this negative region.

Thus, for $x\in [0,\frac{3}{2}]$, $|2x^2-3x|=-(2x^2-3x)=3x-2x^2$. For $x\in [\frac{3}{2},2]$, the parabola is positive, so it remains $2x^2-3x$. We split our integral at $\frac{3}{2}$ to account for this change in behavior.

Calculating the first part:

For the second part:

Adding these together, we find:

Now, let us turn to $I_2={\int }_0^2[x-\frac{1}{2}]dx$. The greatest integer function is a step function that remains constant between integers. We need to find where the 'jumps' occur by setting $x-\frac{1}{2}=k$.

Within our range $[0,2]$, the jumps happen at $x=\frac{1}{2}$ and $x=\frac{3}{2}$. This divides our integral into three intervals: $[0,\frac{1}{2})$, $[\frac{1}{2},\frac{3}{2})$, and $[\frac{3}{2},2]$.

In the first interval, $x-\frac{1}{2}$ is between $-0.5$ and $0$, so the greatest integer is $-1$. In the second, it is between $0$ and $1$, so the value is $0$. In the third, it is between $1$ and $1.5$, so the value is $1$.

The integral becomes:

This is equivalent to the area of rectangles:

We have reached the end. We found $I_1=\frac{19}{12}$ and $I_2=0$.

Adding them together, the final result is:

By breaking the problem down, we turned a terrifying expression into a series of simple, logical steps. Keep this mindset, and there is no problem in the JEE that can stand against you.