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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.

Explore the detailed step-by-step solution to the integral ∫x(2x + 1)^3 dx using substitution and integration by parts. This guide explains the mathematical concepts and formulas involved, including the use of u-substitution and the fundamental theorem of calculus.

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