Animated Solution for Mathematics - Definite Integration: The value of ${\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$, where $[t]$ denotes the greatest integer function, is :

To solve the integral $I={\int }_0^{2\pi }[\sin 2x(1+\cos 3x)]dx$, we first define the function inside the greatest integer brackets as:

The goal is to evaluate the integral of the floor function $[g(x)]$ over the interval $[0,2\pi ]$.

The limits of integration, $0$ to $2\pi$, suggest the use of the King's Property, which states:

Applying this to our function $g(x)$ with $a=2\pi$, we substitute $x$ with $2\pi -x$:

Combining these results, we find the symmetry of the function:

We now express the integral $I$ in two ways:

Adding these two equations yields:

We utilize the property of the greatest integer function, which states that $[t]+[-t]=-1$ for all $t otin \mathbb{Z}$. Since the set of points where $g(x)$ is an integer has measure zero, we can treat the integrand as $-1$ almost everywhere:

Dividing by $2$, we arrive at the final result: