Trapezoidal Rule for Integration

I = \sum_{i = 1}^{n} \frac{h (f (x_{i}) + f (x_{1 + i}))}{2}

Split sum

I = \sum_{i = 1}^{n} \frac{h f (x_{i})}{2} + \sum_{i = 1}^{n} \frac{h f (x_{1 + i})}{2}

Adjust limits

I = \sum_{i = 0}^{-1 + n} \frac{h f (x_{i})}{2} + \sum_{i = 1}^{n} \frac{h f (x_{i})}{2}

Peel terms

I = 2 \sum_{i = 1}^{-1 + n} \frac{h f (x_{i})}{2} + \frac{h f (x_{0})}{2} + \frac{h f (x_{n})}{2}