SCIE 001 MATHEMATICS ASSIGNMENT 10 (Due Jan. 30, 2013 )
There are two parts to this assignment. The first part is on WeBWorK and is due by 10:00 am on Wed.
Jan. 30. The second part consists of the questions on this page, and is due before 10:00 am on Wed. Jan. 30.
For these questions, you are expected to provide full solutions with complete arguments and justifications.
You will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must
be typeset or very neatly written. They must be stapled, with your name and student number at the top of
each page.
1. (a) Use the reduction formula in Example 6, p. 467 to show that if n ≥ 2 is an integer, then
Z
π/2
sinn x dx =
0
n−1
n
Z
π/2
sinn−2 x dx.
0
(b) Prove that, for any integer n ≥ 1, we have
Z
π/2
sin2n+1 x dx =
0
2 · 4 · 6 · · · · · 2n
.
3 · 5 · 7 · · · · · (2n + 1)
(Use the same method of proof, called induction, as for Assignment 8, Question 2(a) i. and ii. – this
has nothing to do with magnetic fields.)
(c) Prove (using induction again) that, for any integer n ≥ 1, we have
Z
π/2
sin2n x dx =
0
1 · 3 · 5 · · · · · (2n − 1) π
.
2 · 4 · 6 · · · · · 2n
2
R π/2
(d) Let In = 0 sinn x dx for integers n ≥ 1, and show that I2n+2 ≤ I2n+1 ≤ I2n for any integer n ≥ 1.
(e) Show that, for any integer n ≥ 1, we have
I2n+2
2n + 1
=
,
I2n
2n + 2
and then
and finally deduce that
lim
n→∞
(f) Prove that
lim
n→∞
2n + 1
I2n+1
≤
≤ 1,
2n + 2
I2n
I2n+1
= 1.
I2n
2 2 4 4 6 6
2n
2n
π
· · · · · ·····
·
= .
1 3 3 5 5 7
2n − 1 2n + 1
2
(This formula is usually written as an infinite product
π
2 2 4 4 6 6
= · · · · · · ...,
2
1 3 3 5 5 7
and is called Wallis’ product or the Wallis product.)
2. Show that
Z
b
sin x
cos a cos b
dx =
−
+
x
a
b
a
R∞
for 0 < a < b < ∞, and deduce that 0 (sin x)/x dx exists.
Z
b
a
cos x
dx
x2