Integrating a polynomial over a fixed interval is usually very straightforward. However, I can't seem to get very far with an $n$-th degree polynomial:
$\int_a^b \bigg(\sum_{i=0}^n q_i\,x^{n-i}\bigg)dx = \bigg[\sum_{i=0}^n \frac{q_i}{1+n-i}\,x^{1+n-i}\bigg]_a^b \\ = \bigg(\sum_{i=0}^n \frac{q_i}{1+n-i}\,b^{1+n-i}\bigg) - \bigg(\sum_{i=0}^n \frac{q_i}{1+n-i}\,a^{1+n-i}\bigg) \\ = \sum_{i=0}^n \bigg(\frac{q_i}{1+n-i}\,b^{1+n-i} - \frac{q_i}{1+n-i}\,a^{1+n-i}\bigg)$
Am I doing this correctly? If so, what comes next? If not, what am I doing wrong?