The problem asks us to evaluate the definite integral $\int (2x+1)^7 dx$.

1. Problem Description

The problem asks us to evaluate the definite integral $\int (2x+1)^7 dx$.

2. Solution Steps

We can use the substitution method to solve this integral.

Let $u = 2x+1$. Then, the derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 2$.

So, $du = 2dx$, and $dx = \frac{1}{2} du$.

Now, we can rewrite the integral in terms of $u$:

$\int (2x+1)^7 dx = \int u^7 \cdot \frac{1}{2} du = \frac{1}{2} \int u^7 du$

Now, we can use the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.

Applying the power rule to our integral, we get:

$\frac{1}{2} \int u^7 du = \frac{1}{2} \cdot \frac{u^{7+1}}{7+1} + C = \frac{1}{2} \cdot \frac{u^8}{8} + C = \frac{u^8}{16} + C$

Now, we substitute back $u = 2x+1$ to express the result in terms of $x$:

$\frac{(2x+1)^8}{16} + C$

3. Final Answer

$\frac{(2x+1)^8}{16} + C$