Write down the integral of the following
cos 2x

Answer

To find the integral of \( \cos(2x) \, dx \), we can follow these steps:

### Step-by-Step Solution:

1. **Substitution**: Let \( t = 2x \). Then, differentiate both sides with respect to \( x \):
\[ dt = 2 \, dx \quad \text{or} \quad dx = \frac{dt}{2} \]

2. **Rewrite the Integral**: Substitute \( t \) and \( dx \) into the integral:
\[ \int \cos(2x) \, dx = \int \cos(t) \cdot \frac{dt}{2} \]

3. **Factor Out the Constant**: The integral can be simplified by factoring out \( \frac{1}{2} \):
\[ = \frac{1}{2} \int \cos(t) \, dt \]

4. **Integrate**: The integral of \( \cos(t) \) is \( \sin(t) \):
\[ = \frac{1}{2} \sin(t) + C \]

5. **Back Substitute**: Replace \( t \) with \( 2x \):
\[ = \frac{1}{2} \sin(2x) + C \]

### Final Answer:
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \]