Animated Solution for Mathematics - Definite Integration: If $24{\int }_0^{\frac{\pi }{4}}(\sin |4x-\frac{\pi }{12}|+[2\sin x])dx=2\pi +\alpha ,$ where [ ] denotes the greatest integer function, then a is equal to

Welcome, future engineer. Today, we are going to dissect a problem that might look like a chaotic mess of symbols, but is actually a beautifully choreographed dance between two distinct mathematical worlds: the discrete and the continuous.

We are tasked with evaluating the integral:

When you see an integral like this, your first instinct might be panic. However, we can utilize the linearity of integrals to split this into two manageable beasts:

The Discrete World of the GIF

Let us start with $I_2$, the Greatest Integer Function. The function $[2\sin x]$ is a step function that only changes value when the input $2\sin x$ hits an integer.

In our interval $x\in [0,\frac{\pi }{4}]$, the value of $2\sin x$ ranges from $0$ to $\sqrt{2}\approx 1.414$. The only integer it crosses is $1$.

Setting $2\sin x=1$ gives $\sin x=\frac{1}{2}$, which occurs at $x=\frac{\pi }{6}$. This is our critical point.

For $x\in [0,\frac{\pi }{6})$, $2\sin x<1$, so $[2\sin x]=0$. For $x\in [\frac{\pi }{6},\frac{\pi }{4}]$, $1\le 2\sin x<2$, so $[2\sin x]=1$.

The integral $I_2$ simplifies to:

The Continuous World of Absolute Values

Now, let us tackle $I_1={\int }_0^{\frac{\pi }{4}}\sin |4x-\frac{\pi }{12}|dx$. To eliminate the absolute value, we use the substitution $t=4x-\frac{\pi }{12}$, which implies $dt=4dx$ or $dx=\frac{dt}{4}$.

The limits change as follows: when $x=0$, $t=-\frac{\pi }{12}$; when $x=\frac{\pi }{4}$, $t=\frac{11\pi }{12}$. The integral becomes:

Splitting the integral at $t=0$ where $|t|$ changes definition:

Evaluating these components:

The Grand Finale

When we add these two parts together, the cosine terms vanish:

Finally, we combine $I_1$ and $I_2$:

The question asks for $24I$:

Comparing this to the form $2\pi +\alpha$, we find $\alpha =12$.