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The integral of the expression $(2x+1)^3$ can be computed as follows:

To solve this integral, we can utilize the power rule of integration, which states that the integral of $x^n$ is given by:

We will apply this rule to each term in the expanded form of $(2x+1)^3$. Let’s first expand the expression:

$(2x+1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + 1^3 = 8x^3 + 12x^2 + 6x + 1$

Now, we can integrate each term separately using the power rule:

1. For the term $8x^3$:

   $$\int 8x^3 \, dx = \frac{8}{4} x^4 + C = 2x^4 + C$$

2. For the term $12x^2$:

   $$\int 12x^2 \, dx = \frac{12}{3} x^3 + C = 4x^3 + C$$

3. For the term $6x$:

   $$\int 6x \, dx = \frac{6}{2} x^2 + C = 3x^2 + C$$

4. For the constant term $1$:

   $$\int 1 \, dx = x + C$$

Now, we can combine these integrated terms:

Next, we can simplify the integral by factoring out a common factor of $(2x+1)$:

Using a substitution where $u = 2x + 1$, we have $du = 2 \, dx$, or equivalently, $dx = \frac{1}{2} du$:

Now, we can apply the power rule to integrate $u^2$:

Finally, we multiply through by $\frac{4}{2}$ to adjust the coefficient:

Thus, we conclude that the integral of $(2x+1)^3$ is: