Calculus: Analytic Geometry and Calculus, with Vectors

8.2 Trigonometric integrands 457

which is correct whenever p > -1 and q > -1. As usual, 0! = 1, 1! = 1,
2! = 1.2, 3! = 1.2.3, 41 = 1.2.3.4, etcetera. The values of x! when x is not an
integer are more esoteric, but x! > 0 when x > -1. Put p = q = 0 and dis-
cover that(--cT)! = a. Show that the formula is correct when p = q = 1.
put p = q = 2 and discover that ()! = V/2. Put p = 2x - 1 and q = 1
and use the result to prove that x! is the product of x and (x - 1)!.
7 With an assist from (8.144), which shows that

r

f01(1 + 6)2

dO <f01 (sin0)2dO < f 1 1 do,

the middle integral being a Riemann-Cauchy integral because the integrand is

undefined when 0 = 0, show that

i 1 sin B 2

f0 ( 9 ) d851.

Then prove the first of the inequalities

sin B 2 sin6)2

0<_fl

°° (sin

)do51, V<f0 w ( ) d8<2

and use it to obtain the second.

8 Let a and b be constants for which a = 0 and b = 1 or a = 1 and b = 0

and let x > 0. Show that

-15acosx+bsin xS1.

Replace x by t and integrate over the interval 0 <= t x to obtain

-xS asinx-b(cosx-1) Sx.

Replace x by t and integrate over the interval 0 5 t =< x to obtain

• X2 -a(cos x - 1) - b(sin x - x) 2

Repeat the process to obtain

3

23

a^4

x-x+3i)<4i

2 4 6

-Si5a(sinx-x+3$-b(cosx--+Zi-4!)S!

Repeat the process two more times. With or without more attention to details,

jump to the conclusion that

xs x6 x7 x2n+i 1x1 2n+3

sinx- x31+ j i+ 1I 2n 3

x2 x4 xe x2n

l

IxI1n+2

Icosx-(1-i+4i-6i{ ... h(2n))I<(2n+2)1