The problem asks us to find the antiderivatives of a list of functions. In other words, we need to find indefinite integrals of the given expressions. Here, I will solve part a: $\int x^7 dx$.

1. Problem Description

The problem asks us to find the antiderivatives of a list of functions. In other words, we need to find indefinite integrals of the given expressions. Here, I will solve part a: $\int x^7 dx$.

2. Solution Steps

To find the antiderivative of $x^7$, we will use the power rule for integration:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.

In our case, $n = 7$. Applying the power rule, we have:

$\int x^7 dx = \frac{x^{7+1}}{7+1} + C = \frac{x^8}{8} + C$.

3. Final Answer

The antiderivative of $x^7$ is $\frac{x^8}{8} + C$.