Calculus: Analytic Geometry and Calculus, with Vectors

(^470) Trigonometric functions
a sense in which this whole business is utterly silly, but it really is a
convenience to imagine that dx, u'(t), and dt are three numbers for which
dx = u(t) dt. When x = u(t), we can differentiate to get
dx
= u'(t),
and the pretense that dx is the quotient of two numbers then enables us
to multiply by dt to get the formula dx = u'(t) dt that tells us to replace
dx by u(t) dt. It is possible to try to say more about these matters but,
for present purposes, the important thing is that we do not forget the
factor u'(t).
Since it is not always convenient to introduce a new symbol, such as
u, for the function which carries t into x, we restate the theorem in the
following form which can be ignored by those who prefer the first version.

Theorem 8.433 If

(8.434) F(x) = f f (x) dx

and if x is a differentiable function whose range lies in the domain of f,

then

(8.435) F(x(t)) =ff(x(t))x'(t) dt

or

(8.436) F(x(t)) =ff(x(t)) dt

(8.437) F(x) =ff(x)dx.

We give further attention to changes of variables in integrals by prov-

ing the following very useful variant of Theorem 8.433 which involves

Riemann integrals.

Theorem 8.44 If f is continuous over the interval a < x 5 b and if is

is a function which has a continuous derivative and is such that u(t) increases

from a to b as t increases (or decreases) from a to 0, then

(8.441)

fab

f(x) dx = f

8

f(x(t))x'(t) dt.

To prove this theorem, let F be a function whose derivative is fso that

f f (x) dx = F(x) + c and

(8.442) fab f(x) dx=F(x)]a = F(b) - F(a).

The chain formula for derivatives then implies that

(8.443)

f

a

f(x(t))u'(t) dt = F(u(t)) ]a= F(u(f4)) - F(u(a)).