Calculus: Analytic Geometry and Calculus, with Vectors

454 Trigonometric functions

Instead of inviting attention to problems of this nature, we present prob-

lems more likely to promote scientific competence.

Table 8.28

Jsinududx=-cosu+c

f tanudxdx=logIsecul+c

f

secudxdx=log Isecu+tanul+c

f

dduxdx

n+1+c'f

uddu

Jcosudxdx=sinu+c

Jcot u du dx = log sin ul + c

I csc is du dx = log Icsc u - cot ul + c

x

du

+ c, f eud dr= eu}c.

dx

Problems 8.29

1 Make all of the calculations necessary to show that

(a)

(c)

(e)

(g)

Iosin x dx = 2

b

lim J sin wx dx = 0a

1 r

lim

J

sin2 wt dt =

xdx = log 2

(b)

fox"

cos x dx = 1

x

?/2.

(h) fo tan x dx

(d) lim f cos wx dx = 0

(f) lim

1

• fo wt dt =
  -m x o

x

= 00

2 Recall that, when is is a differentiable function of x and is 0 0,

(1) dxlog Jul u dx'

the absolute-value signs, which are superfluous when u > 0, need not bother us.

Supposing that x is not an odd multiple of a/2 and that

f(x) = log sec x + tan xl, g(x) = log I tan 0 + 4) I,

show that f' (x) = sec x and g'(x) = sec x.

Remark: This proves that the two formulas

A4)

fsee x dx = log sec x + tan xl + Cl

f secxdx=logItan(2+7rf see x dx = log I tan 0 )I+s

are both correct. Some integral tables contain both of them. These things

imply that, over each interval containing no odd multiple of 7r/2, f(x)- g(x)