Q.Evaluate $\displaystyle\int_{1}^{4} (2x+1)\,dx$ directly, and again by splitting the interval at $x=2$ using the additivity property. Confirm both methods agree.

The additivity property says an integral over $[a,b]$ equals the sum of the integrals over $[a,c]$ and $[c,b]$ for any point $c$ between them; both routes must give the same number.

Antiderivative: $F(x)=x^{2}+x$.

Method 1 — direct evaluation:

Method 2 — split at $c=2$ using additivity:

$\int_{1}^{2}(2x+1)\,dx + \int_{2}^{4}(2x+1)\,dx = 4+14 = 18$ …

Splitting the interval but then adding $F(1)-F(2)$ and $F(2)-F(4)$ (limits reversed within each piece) instead of $F(2)-F(1)$ and $F(4)-F(2)$ — each sub-integral …