Evaluate the following :
`int (e^x +3 cos x-4x^3+2) dx.`

Answer

To evaluate the integral \( \int (e^x + 3 \cos x - 4x^3 + 2) \, dx \), we will integrate each term separately.

### Step-by-step Solution:

1. **Integrate \( e^x \)**:
\[ \int e^x \, dx = e^x \]

2. **Integrate \( 3 \cos x \)**:
\[ \int 3 \cos x \, dx = 3 \sin x \]

3. **Integrate \( -4x^3 \)**:
\[ \int -4x^3 \, dx = -4 \cdot \frac{x^4}{4} = -x^4 \]

4. **Integrate \( 2 \)**:
\[ \int 2 \, dx = 2x \]

5. **Combine all the results**:
Now, we sum all the integrated terms:
\[ \int (e^x + 3 \cos x - 4x^3 + 2) \, dx = e^x + 3 \sin x - x^4 + 2x + C \]
where \( C \) is the constant of integration.

### Final Answer:
\[ \int (e^x + 3 \cos x - 4x^3 + 2) \, dx = e^x + 3 \sin x - x^4 + 2x + C \]