`int_(0)^(16n^(2)//pi) "cos" (pi)/(2)[(xpi)/(n)] dx` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the integral \( \int_{0}^{\frac{16n^2}{\pi}} \cos\left(\frac{\pi}{2} \cdot \frac{x\pi}{n}\right) dx \), we will follow these steps:

### Step 1: Simplify the Integral
We can rewrite the integral as:
\[ \int_{0}^{\frac{16n^2}{\pi}} \cos\left(\frac{\pi^2 x}{2n}\right) dx \]

### Step 2: Change of Variables
Let \( t = \frac{\pi^2 x}{2n} \). Then, we differentiate:
\[ dt = \frac{\pi^2}{2n} dx \quad \Rightarrow \quad dx = \frac{2n}{\pi^2} dt \]
Now we need to change the limits of integration. When \( x = 0 \), \( t = 0 \), and when \( x = \frac{16n^2}{\pi} \):
\[ t = \frac{\pi^2 \cdot \frac{16n^2}{\pi}}{2n} = \frac{16\pi n}{2} = 8\pi n \]
So the integral becomes:
\[ \int_{0}^{8\pi n} \cos(t) \cdot \frac{2n}{\pi^2} dt \]

### Step 3: Factor Out Constants
We can factor out the constant \( \frac{2n}{\pi^2} \):
\[ \frac{2n}{\pi^2} \int_{0}^{8\pi n} \cos(t) dt \]

### Step 4: Evaluate the Integral
The integral of \( \cos(t) \) is \( \sin(t) \):
\[ \int \cos(t) dt = \sin(t) \]
Thus,
\[ \int_{0}^{8\pi n} \cos(t) dt = \sin(8\pi n) - \sin(0) = \sin(8\pi n) \]
Since \( \sin(8\pi n) = 0 \) for any integer \( n \), we have:
\[ \int_{0}^{8\pi n} \cos(t) dt = 0 \]

### Step 5: Final Result
Putting it all together:
\[ \frac{2n}{\pi^2} \cdot 0 = 0 \]
Thus, the value of the original integral is:
\[ \int_{0}^{\frac{16n^2}{\pi}} \cos\left(\frac{\pi}{2} \cdot \frac{x\pi}{n}\right) dx = 0 \]

### Conclusion
The final answer is:
\[ \boxed{0} \]