`intsin^(2) n x dx`

Answer

To solve the integral \( \int \sin^2(nx) \, dx \), we will use a trigonometric identity to simplify the expression. Here’s a step-by-step solution:

### Step 1: Use the Trigonometric Identity
We know that:
\[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \]
Applying this identity to \( \sin^2(nx) \):
\[ \sin^2(nx) = \frac{1 - \cos(2nx)}{2} \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \sin^2(nx) \, dx = \int \frac{1 - \cos(2nx)}{2} \, dx \]
This can be separated into two integrals:
\[ = \frac{1}{2} \int (1 - \cos(2nx)) \, dx = \frac{1}{2} \left( \int 1 \, dx - \int \cos(2nx) \, dx \right) \]

### Step 3: Integrate Each Term
Now we will integrate each term separately:
1. The integral of \( 1 \) is:
\[ \int 1 \, dx = x \]
2. The integral of \( \cos(2nx) \) is:
\[ \int \cos(2nx) \, dx = \frac{\sin(2nx)}{2n} \]

### Step 4: Combine the Results
Now substituting back into our expression:
\[ \int \sin^2(nx) \, dx = \frac{1}{2} \left( x - \frac{\sin(2nx)}{2n} \right) \]
This simplifies to:
\[ = \frac{x}{2} - \frac{\sin(2nx)}{4n} + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result is:
\[ \int \sin^2(nx) \, dx = \frac{x}{2} - \frac{\sin(2nx)}{4n} + C \]

---