The integral given is:

$\int x (2x + 1)^3 \, dx$

To solve this, we will use substitution and integration by parts.

Step 1: Substitution

Let: $u = 2x + 1$ Then: $\frac{du}{dx} = 2 \quad \Rightarrow \quad dx = \frac{du}{2}$ Also, we have: $x = \frac{u - 1}{2}$

Step 2: Rewriting the integral

Substitute $x = \frac{u - 1}{2}$ and $dx = \frac{du}{2}$ into the integral: $\int x (2x + 1)^3 \, dx = \int \frac{u - 1}{2} u^3 \frac{du}{2}$

Simplifying: $\int \frac{(u - 1) u^3}{4} \, du = \frac{1}{4} \int (u^4 - u^3) \, du$

Step 3: Integrating term by term

Now, integrate $u^4$ and $u^3$ term by term: $\frac{1}{4} \left( \frac{u^5}{5} - \frac{u^4}{4} \right) + C$

Step 4: Substituting back for $u$

Since $u = 2x + 1$, substitute this back into the result: $\frac{1}{4} \left( \frac{(2x + 1)^5}{5} - \frac{(2x + 1)^4}{4} \right) + C$

This is the final answer.

Would you like any further clarification or details?

Related Questions:

1. How does substitution help simplify integrals?
2. What other methods can be used to solve integrals involving polynomials?
3. What is the significance of the constant of integration $C$?
4. How would the result change if we had a different power of $(2x + 1)$?
5. Can this integral be solved using numerical methods?

Tip:

Substitution is particularly useful when the integrand includes a composite function that simplifies into a basic form after substitution.