Calculus: Analytic Geometry and Calculus, with Vectors

9.5 Integration by parts 523

and show that multiplying the numerators and denominators of the right mem-
bers of (2) and (3) by theleft member of (4) gives the formulas

(5) or/2sin2nxdx =22nIn !2f 2

r/2 22nn!n!

sin2n+1 x dx =

1

(6) o (2n)! 2n -

One reason for interest in these things lies in the fact that the number

(2n)!

22' n!nl

is the probability of finding exactly n heads and exactly n tails when 2n coins are

tossed. Remark: We embark on a little excursion to see that these formulas

have startling consequences. When 0 < x < 2r/2, we have 0 < sin x < 1, so

(7) 0 < sin2n+l x < sin2n x < sin2n-1 x < 1

and hence

,r/2 r/2 x/2

(8) 0< fo sin2n+1 x dx < fo in2n x dx < f^0 sin2n-1 x dx.

Putting p = 2n + 1 in (1) gives the formula

(9) f

a/2 1 /2

o sin2n-1 x dx = (1 +2n/fo sin2n+1 x dx.

r

It follows from (8) and (9) that there is a number Bn for which 0 < 0n < 1 and

(10) for/2 sin 2nx dx =(1 + 2-n)

for/2

sin2n+i x dx.

:Multiplying the members of (5) and (6) gives

(11) Lfor/2 sin2n x dx ILrfor/2 sin2n+1 x dx =

4i1(1+2n!

Multiplying the members of (10) and (11) leads to the formula

/2 1 - V

(12)

for

sin2n x dx = 1

Bn

• 2n -}- 12

Substituting this in (5) gives the formula

(2n)! 1 - 0n^1

(13) 22nnln! =^1 - 2n + 1

Since 0 < On < 1, the first factor in the right member is near 1 when n islarge