Calculus: Analytic Geometry and Calculus, with Vectors

8.4 Integration by trigonometric and other substitutions 469

17 Try to understand and even prove the statement that each of the operators

Dl and Il (of the two preceding problems) is the inverse of the other.

8.4 Integration by trigonometric and other substitutions We

begin with a statement of the fact that there is an elementary function

F whose derivative with respect to x is a2 - x2. Letting F denote

one such function, we try to learn about F by writing

(8.41) F(x) = f a2 - x2 dx

and searching for an idea. Once upon a time somebody discovered that if

we substitute x = a sin 0, then the integrand will become a2 - a2 sin2 6

or a2 cos2 0, and this is a cos 0 if a cos 0 > 0. Thus a trigonometric

substitution removes the radical and leaves ±a cos 6, but we can still

be unsure of the meaning of f cos 0 dx. Hence we must pause to make

an observation.

The chain rule of Theorem 3.65 tells us that if

(8.42) F'(x) = f(x)

and if u is a differentiable function whose range lies in the domain of F,

then

(8.421) dtF(u(t))= F'(u(t))u'(t)

= f(u(t))u'(t).

This gives the following important substitution theorem which shows

how to replace x by u(t) in integrals.

Theorem 8.43 If

(8.431) F(x) = f f (x) dx

andsif u is a differentiable function whose range lies in the domain of f, then

(8.432) F(u(t)) = ff(u(t))u'(t) dt.

This theorem is used so often that it is worthwhile to be able to get
from the right side of (8.431) to the right side of (8.432) in a purely
formal way without thinking about the way in which the theorem was
proved and the meanings of the formulas. We replace the old integrand
f(x) by the new integrand f(u(t))u'(t) and the old dx by the new dt.

This seems like a simple ritual, but the factor u'(t) might be forgotten

when problems are being solved. It seems to be safer and easier to think

of f(x) being replaced by f(u(t)) and dx being replaced by u'(t) dt. If

we follow historical precedents, we become carried away by our own

enthusiasm and try to eliminate all possibility of overlooking the factor

u'(t) by creating the fiction that dx is a number (a bunch of bananas

would serve the same purpose) and u'(t) dt is the same thing. There is