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

1. Problem Description

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

2. Solution Steps

We can solve this integral using u-substitution.

Let $u = 2x + 1$.

Then, $\frac{du}{dx} = 2$, which means $dx = \frac{1}{2} du$.

Substituting $u$ and $dx$ into the integral, we have:

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

Now we can use the power rule for integration:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

Applying the power rule to our integral, we get:

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

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

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

3. Final Answer

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