`int cos (2x +1) dx`

Answer

To solve the integral \( \int \cos(2x + 1) \, dx \), we will use the technique of substitution and the basic integration rule for cosine.

### Step-by-Step Solution:

1. **Identify the inner function**:
Let \( u = 2x + 1 \). Then, we need to find \( du \).

\[ du = \frac{d}{dx}(2x + 1) \, dx = 2 \, dx \implies dx = \frac{du}{2} \]

2. **Substitute in the integral**:
Now, substitute \( u \) and \( dx \) into the integral:

\[ \int \cos(2x + 1) \, dx = \int \cos(u) \cdot \frac{du}{2} \]

3. **Factor out the constant**:
We can take the constant \( \frac{1}{2} \) outside the integral:

\[ = \frac{1}{2} \int \cos(u) \, du \]

4. **Integrate \( \cos(u) \)**:
The integral of \( \cos(u) \) is \( \sin(u) \):

\[ = \frac{1}{2} \sin(u) + C \]

5. **Substitute back for \( u \)**:
Now, substitute back \( u = 2x + 1 \):

\[ = \frac{1}{2} \sin(2x + 1) + C \]

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