1549901369-Elements_of_Real_Analysis__Denlinger_

7.6 The Fundamental Theorem of Calculus 409

A few words about this notation are in order. The justification for using the

"integral sign" for an antiderivative off comes from FTC-II: on any interval

[a, b] where f is continuous, fax f is one such antiderivative. The reason for

using x and dx in the notation J f (x )dx lies in the suggestive power of Leibniz's

notation for derivatives and differentials. In this notation, whenever y = f(x)

is differentiable we denote its derivative by ~ or d~~), and its differential by

dy = ~dx = f'(x)dx, or

df = df(x) = d~~) dx = f'(x)dx.

From this perspective, to find J f(x)dx means to find a function F whose
differential is f(x)dx.
In elementary calculus, we learn to exploit differential notation in finding
indefinite integrals. For example, to find J tan^2 x sec^2 x dx we notice that if we
let u = tan x then du = sec^2 xdx, so

Jtan^2 xsec^2 xdx = J u^2 du = 1u^3 +C=1 tan^3 x + C.

We often call this the method of "u-substitution,'' or "change of variables."
In this method, to find J f(x)dx, we look for a differentiable function u and an
integrable function g such that

f(x)dx = g(u)u'(x)dx = g(u)du. Then

J f(x)dx = J g(u)du.

Of course, this procedure is most helpful if J g(u)du is "easier" to find
than the original J f(x)dx. Finally, note that in changing one differential into
another we are using the chain rule. In the following theorem we give a formal
proof of the validity of this procedure.

Theorem 7.6.13 (Change of Variables, or Substitution) Suppose u is
differentiable on [a, b], u' is integrable on [a, b], and g is continuous on u[a, b].
Then (go u)u' is integrable on [a, b] and

l

b 1 u(b)

(go u)u' = g.

a u(a)

In notation more familiar from elementary calculus,

l

b 1 u(b)

(g(u(x))u'(x) dx = g(u)du.

a u(a)

Proof. Suppose u is differentiable on [a, b], u' is integrable on [a, b], and
g is continuous on u[a,b]. Let c = u(a) and d = u(b). Since u is continuous