The value of the integral ` int_(0)^(pi//2) sin 2n x cot x dx,` where n is a positive integer, is

Options

A. `(pi)/(2)`
B. `-pi`
C. `pi`
D. none of these

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \sin(2nx) \cot(x) \, dx \), where \( n \) is a positive integer, we will follow these steps:

### Step 1: Rewrite the integral
We know that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Therefore, we can rewrite the integral as:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin(2nx) \frac{\cos(x)}{\sin(x)} \, dx \]
This simplifies to:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin(2nx) \cos(x) \, \frac{dx}{\sin(x)} \]

### Step 2: Use the identity for \( \sin(2nx) \)
Using the identity \( \sin(2nx) = 2 \sin(nx) \cos(nx) \), we can express the integral as:
\[ I = \int_{0}^{\frac{\pi}{2}} 2 \sin(nx) \cos(nx) \cos(x) \, \frac{dx}{\sin(x)} \]

### Step 3: Simplify the integral
Now, we can factor out the constant:
\[ I = 2 \int_{0}^{\frac{\pi}{2}} \sin(nx) \cos(nx) \cos(x) \, \frac{dx}{\sin(x)} \]

### Step 4: Change of variable
To simplify the integral further, we can use the formula:
\[ \int_{0}^{\frac{\pi}{2}} \sin(2nx) \cot(x) \, dx = \sum_{k=1}^{n} \frac{1}{k} \]
This gives us a direct way to evaluate the integral without needing to compute it step by step.

### Step 5: Evaluate the integral
Using the formula derived, we find:
\[ I = \sum_{k=1}^{n} \frac{1}{k} \]
This is the harmonic sum of the first \( n \) positive integers.

### Conclusion
Thus, the value of the integral \( \int_{0}^{\frac{\pi}{2}} \sin(2nx) \cot(x) \, dx \) is:
\[ \boxed{H_n} \]
where \( H_n \) is the \( n \)-th harmonic number.