Definite integrals and the fundamental theorem

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Find a primitive and evaluate upper minus lower endpoint. This evaluation requires the theorem conditions; singularities require improper integrals. Transform bounds under substitution.

Step-by-step example

1. Find a primitive.

   $$\int_0^2x^2\,dx,\quad F(x)=x^3/3$$

2. Evaluate both endpoints.

   $$F(2)-F(0)=8/3-0$$

3. The definite result has no C.

   $$I=8/3$$