Calculus: Analytic Geometry and Calculus, with Vectors

8.2 Trigonometric integrands 451

the numerator and denominator of (sec x)/1 by the implausible factor
sec x + tan x. We dispose of csc x as rapidly as possible by writing

(8.236)

J

csc x dx fcsc x } cot x(-csc2 x - csc x cot x) dx

log Icsc x + cot XI + C.

When 0 < x < it as in Figure 8.234, we can omit the absolute-value

signs. The chain extensions of the six formulas are placed in a table at
the end of this section where they are most available for reference. The

formulas

(8.24)

1

sin ax dx a

J

sin ax a dx = - 1 cos ax + c

(8.241) fcos ax dx =a fcos ax a dx = a sin ax + c,

which hold when a 0, are by far the most important applications of

the chain extensions.

We now consider some of the more or less important integrals that can

be evaluated in terms of elementary functions. Adding and subtracting

the two elementary formulas

(8.25) cost 0 + sin2 8 = 1

(8.251) cost B - sin2 0 = cos 20

gives the two formulas

(8.252) cost 0 =

1+cos20

2 -,

1 - cos 20

sin2 0 = 2

that are so useful that we should either learn them or be able to work them

out very quickly. Moreover, these formulas should come to mind when

we are asked to evaluate the first integrals in the formulas

(8.253) fcos2axdx=

f 1 + s 2axdx

= 2[x+_fcos2ax(2a)dx]

=2Lx+-

sin2 ax

s dx

fcos2ax(2a)dx]

=2rx-1sin2ax,+c.