`int cos (2x +1) dx`

Answer

To solve the integral \( \int \cos(2x + 1) \, dx \), we will follow these steps:

### Step 1: Set up the integral
Let:
\[ I = \int \cos(2x + 1) \, dx \]

### Step 2: Use substitution
We will use the substitution:
\[ t = 2x + 1 \]
Now, differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = 2 \implies dt = 2 \, dx \implies dx = \frac{dt}{2} \]

### Step 3: Substitute in the integral
Substituting \( t \) and \( dx \) into the integral gives:
\[ I = \int \cos(t) \cdot \frac{dt}{2} \]
This can be simplified to:
\[ I = \frac{1}{2} \int \cos(t) \, dt \]

### Step 4: Integrate
Now, we can integrate \( \cos(t) \):
\[ \int \cos(t) \, dt = \sin(t) \]
Thus, we have:
\[ I = \frac{1}{2} \sin(t) + C \]

### Step 5: Substitute back for \( t \)
Now, substitute back \( t = 2x + 1 \):
\[ I = \frac{1}{2} \sin(2x + 1) + C \]

### Final Answer
The final result for the integral is:
\[ \int \cos(2x + 1) \, dx = \frac{1}{2} \sin(2x + 1) + C \]

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