To solve the integral

$I = \int \dfrac{3x^3 - 5x^2 + 10x - 3}{3x + 1} \, dx$

we can start by performing polynomial division on the rational function $\dfrac{3x^3 - 5x^2 + 10x - 3}{3x + 1}$.

Step 1: Perform Polynomial Division

We divide $3x^3 - 5x^2 + 10x - 3$ by $3x + 1$.

1. Divide the leading term of the numerator $3x^3$ by the leading term of the denominator $3x$, which gives $x^2$.

2. Multiply $x^2$ by $3x + 1$, giving $3x^3 + x^2$.

3. Subtract $(3x^3 + x^2)$ from $(3x^3 - 5x^2)$, resulting in $-6x^2 + 10x$.

4. Divide $-6x^2$ by $3x$, which gives $-2x$.

5. Multiply $-2x$ by $3x + 1$, giving $-6x^2 - 2x$.

6. Subtract $(-6x^2 - 2x)$ from $(-6x^2 + 10x)$, resulting in $12x - 3$.

7. Divide $12x$ by $3x$, which gives $4$.

8. Multiply $4$ by $3x + 1$, giving $12x + 4$.

9. Subtract $(12x + 4)$ from $(12x - 3)$, resulting in $-7$.

Thus, the polynomial division gives us:

$\dfrac{3x^3 - 5x^2 + 10x - 3}{3x + 1} = x^2 - 2x + 4 - \dfrac{7}{3x + 1}$

Step 2: Integrate Each Term

Now we integrate term by term:

1. $\int (x^2 - 2x + 4) \, dx = \dfrac{x^3}{3} - x^2 + 4x + C_1$

2. $\int -\dfrac{7}{3x + 1} \, dx$

   We can use substitution for this integral. Let $u = 3x + 1$, so $du = 3 dx$. This gives:

   $\int -\dfrac{7}{3x + 1} \, dx = -\dfrac{7}{3} \int \dfrac{1}{u} \, du = -\dfrac{7}{3} \ln |u| + C_2 = -\dfrac{7}{3} \ln |3x + 1| + C_2$

Step 3: Combine Results

Combining all terms, we get the final result:

$I = \dfrac{x^3}{3} - x^2 + 4x - \dfrac{7}{3} \ln |3x + 1| + C$

where $C = C_1 + C_2$ is the constant of integration.

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Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How does polynomial long division work, and why is it useful for integration?
2. Can we apply synthetic division for polynomials like these?
3. What is the role of substitution in integrals involving logarithms?
4. How would the integral change if we had a different denominator, such as $3x + 2$?
5. Can you explain partial fraction decomposition and when it's applicable?

Tip: When dealing with rational functions, always consider polynomial division or partial fraction decomposition before integrating!