Evaluate : `int(sinx)/(sin 2x)dx`

Answer

To evaluate the integral \( \int \frac{\sin x}{\sin 2x} \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We know that \( \sin 2x = 2 \sin x \cos x \). Therefore, we can rewrite the integral as:
\[ \int \frac{\sin x}{\sin 2x} \, dx = \int \frac{\sin x}{2 \sin x \cos x} \, dx \]

### Step 2: Simplify the expression
Now, we can simplify the fraction:
\[ \int \frac{\sin x}{2 \sin x \cos x} \, dx = \int \frac{1}{2 \cos x} \, dx \]

### Step 3: Factor out the constant
We can factor out the constant \( \frac{1}{2} \):
\[ \int \frac{1}{2 \cos x} \, dx = \frac{1}{2} \int \sec x \, dx \]

### Step 4: Integrate \( \sec x \)
The integral of \( \sec x \) is known to be:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]
Thus, we have:
\[ \frac{1}{2} \int \sec x \, dx = \frac{1}{2} \left( \ln |\sec x + \tan x| + C \right) \]

### Step 5: Write the final answer
Putting it all together, we get:
\[ \int \frac{\sin x}{\sin 2x} \, dx = \frac{1}{2} \ln |\sec x + \tan x| + C \]

### Summary of the steps:
1. Rewrite the integral using the identity for \( \sin 2x \).
2. Simplify the fraction.
3. Factor out the constant.
4. Integrate \( \sec x \).
5. Write the final answer.