**Integration of 3x^3 + 2x^2 + 3x + 1/3(x^2 + 1)**

To integrate the given expression, we will integrate each term separately using the power rule of integration.

1. **Integrating the term 3x^3:**
The power rule of integration states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is a constant.
Applying the power rule, we have:
∫(3x^3) dx = (3/4)x^4 + C1, where C1 is the constant of integration.

2. **Integrating the term 2x^2:**
Using the power rule of integration, we have:
∫(2x^2) dx = (2/3)x^3 + C2, where C2 is the constant of integration.

3. **Integrating the term 3x:**
Again applying the power rule, we have:
∫(3x) dx = (3/2)x^2 + C3, where C3 is the constant of integration.

4. **Integrating the term 1/3(x^2 + 1):**
To integrate this term, we first need to expand it. Using the distributive property, we have:
1/3(x^2 + 1) = 1/3(x^2) + 1/3(1)
= (1/3)x^2 + 1/3
Now, applying the power rule, we integrate each term separately:
- ∫(1/3)x^2 dx = (1/9)x^3 + C4, where C4 is the constant of integration.
- ∫(1/3) dx = (1/3)x + C5, where C5 is the constant of integration.
Combining the two integrals, we have:
∫(1/3(x^2 + 1)) dx = (1/9)x^3 + (1/3)x + C6, where C6 is the constant of integration.

**Final Result:**
Combining all the integrals, we get:
∫(3x^3 + 2x^2 + 3x + 1/3(x^2 + 1)) dx = (3/4)x^4 + (2/3)x^3 + (3/2)x^2 + (1/9)x^3 + (1/3)x + C, where C is the constant of integration.