# Integrating an n-th degree polynomial

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?

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Yes. You are done. What comes next depends on what you want? If you just want to integrate, then you are done. –  user17762 Nov 23 '12 at 19:21