E.2 Integration Foundations

The foundations of integration. The INDEFINITE integral ∫f(x) dx = F(x) + C is the antiderivative — F'(x) = f(x) — with the +C capturing the family of antiderivatives that all share the same derivative (because differentiation discards constants). A single boundary condition (x₀, y₀) PINS DOWN the +C: substitute, set equal to y₀, solve. The DEFINITE integral ∫_a^b f(x) dx returns a NUMBER — the SIGNED area between the curve and the x-axis from x = a to x = b (no +C, since limits eliminate it). Above the axis counts positive; below counts negative. Standard antiderivatives: power rule reverse ∫xⁿ dx = x^(n+1)/(n+1) + C (for n ≠ −1), special case ∫(1/x) dx = ln|x| + C, exponential ∫e^x dx = e^x + C, trig ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C. The REVERSE CHAIN RULE handles composites of the form f(g(x))·g'(x). The FUNDAMENTAL THEOREM OF CALCULUS computes any definite integral via F(b) − F(a). Areas between two curves: ∫_a^b [top − bottom] dx, where a and b are the x-coordinates of the intersection points.