Calculus: Analytic Geometry and Calculus, with Vectors

8.2 Trigonometric integrands 453

is rendered manageable by replacing the factor cos3 x by the last member
of the formula

cos3 x = cost x cos x = (1- sine x) cos x.

The integral in

(8.265) fe-ax sin x dx = - fe00°x(- sin x) dx= -e-ex + c

is evaluated by making the adjustment and compensation necessary to
put it in the form feu(x>u'(x) dx, and opportunities to make such adjust-
ments and compensations should always be observed.
There are reasons why some of the integrals we have evaluated may
be said to be so important that everyone should know how to evaluate
them. There is an old and perhaps honorable tradition that requires
students of the calculus to spend huge amounts of time cultivating "the
technique" of "formal integration." The fact that we live in an age of
electronic computers makes it much more important to learn fundamental
theory than to acquire skill in formal integration. For this reason, the
author requests that teachers join him in avoiding all but the simplest
formal integration problems that are not likely to be encountered by
undergraduates in courses other than calculus courses. "The student"
who has not read dozens of calculus books and does not know what we
are talking about is invited to look at the shiny example

(8.27) I = f sin x tan -x(1+ cos x) dx

of an integral that we shall not expect him to evaluate quickly. A

person who has constructed this problem can easily feel very sure that

the only sensible attack upon the problem lies in setting

(8.271) u(x) = 1 + (cos x);

so, when 0 < x < ir/2,

(8.272) u'(x) _ -(cos x)4(- sin x)

sine x

sin x tan x

cos x

and hence

(8.273) I = -2

J

[u(x)]3f[u'(x)] dx

_ -2 [u(X]I+c= --[l+ cox]V+c.