Calculus: Analytic Geometry and Calculus, with Vectors

450 Trigonometric functions

to an interval containing none of these points, we can use the funda-

mental formula

f [u(x)]-'u'(x) dx = log lu(x)I + c

to obtain

(8.23) tan x dx = -

J

1 (- sin x) dx log 1cos xj -r- c

cos x

=logjsecx1+c.

In most applications of this formula, jxj < it/2, so cos x > 0 and the

absolute-value signs can be omitted. For cot x, a miniature graph of

which appears in Figure 8.232, we suppose that x is confined to an inter-

val containing none of the points x = nar and obtain

(8.231)

J

cot x dx =

J

1 cos x dx = log sin xj + c.

sin x

In most applications of this formula, 0 < x < 7r, so sin x > 0 and the

absolute-value signs can be omitted.

Figure 8.232

y

Figure 8.233

x

y

0

Figure 8.234

A sketch of the graph of y = sec x is most easily obtained by sketching

a graph of y = cos x and estimating reciprocals. As the graph in Figure

8.233 indicates, sec x is undefined when x is an odd integer multiple of

it/2. When we integrate sec x over an interval, we must suppose that

the interval contains none of these points. We can then obtain the

formula

(8.235) fS!_5dx

=fsecx--Itan x

(sect x + sec x tan x) dx

= log (sec x + tan xj + c

provided we happen to know that the result is obtained by multiplying