Formal Integral Definition: f(x) dx = lim (d -> 0) (k=1..n) f(X (k) ) (x (k) - x (k- 1) ) when... a = x 0 < x 1 < x 2 < ... < x n = b d = max (x 1 -x 0 , x 2 -x 1 , ... , x n - x (n-1) ) x (k-1) <= X (k) <= x (k) k = 1, 2, ... , n F '(x) dx = F(b) - F(a) (Fundamental Theorem for integrals of derivatives) a f(x) dx = a f(x) dx (if a is constant) f(x) + g(x) dx = f(x) dx + g(x) dx f(x) dx = f(x) dx | (a b) f(x) dx + f(x) dx = f(x) dx f(u) du/dx dx = f(u) du (integration by substitution)
