6 0. Background
Theorem 0.14. (Fundamental Theorem of Calculus, Second Form) Suppose that
f(x) is Riemann-integrable on the interval [a, b]. Suppose further that F (x) is a
differentiable function on the interval [a, b] such that F (x) = f(x) for a ≤ x ≤ b.
f(x) dx = F (b) − F (a) .
The Second Form of the Fundamental Theorem follows from the First Form if
f is continuous. The point of the Second Form is that it holds under the weaker
assumption that f is Riemann-integrable.
Theorem 0.15. (Integration by parts) Suppose that f is Riemann-integrable on
[a, b], that F is a function such that F (x) = f(x), and that g is a differentiable
function such that g (x) is Riemann-integrable on [a, b]. Then
f(x)g(x) dx = F (b)g(b) − F (a)g(a) −
F (x)g (x) dx .
This follows immediately from the Second Form of the Fundamental Theorem,
in view of the differentiation formula (Fg) = F g + Fg = fg + Fg .
Theorem 0.16. (The triangle inequality for integrals) If f is Riemann-integrable,
f(x) dx ≤
|f(x)| dx .
Theorem 0.17. (Leibniz’s Rule) If f(x, y) and
f(x, y) exist and are continuous
on the closed rectangle a ≤ x ≤ b, c ≤ y ≤ d, then the function
F (x) =
f(x, y) dy
is differentiable for a ≤ x ≤ b, and
F (x) =
f(x, y) dy .
Theorem 0.18. Suppose that f(x) =
fn(x). If the functions fn are differen-
tiable, and if the series
fn(x) is uniformly convergent, then f is differentiable,
f (x) =
Theorem 0.19. (Dominated Convergence) If amn is a double sequence, and An
is such that limm→∞ amn = A exists, and if there is a sequence Mn such that
|amn| ≤ Mn for all m, and
Mn ∞, then