There is a formula for calculating the integral of \(x^{n}\) for any n. It is

\(\int_{a}^{b}x^{n}dx=\frac{1}{n+1}b^{n+1}-\frac{1}{n+1}a^{n+1}\)

So, substituting in n=7n=7, we find that

\(\int_{a}^{b}x^{7}dx=\frac{1}{8}b^{8}-\frac{1}{8}a^{8}\)

Note that this seems a bit too easy so if you haven't gotten to this formula in your class yet and you need to evaluate this integral using the definition or by "guessing" the antiderivative instead of using this rule directly, let me know in a comment and I'll update my answer.