1016-412 RIT, 20092 1
Real Variables II 1016-412
Homework 2
1. Show that if fis bounded and convex on [a, b], then
(b−a)f(a) + f(b)
2≤Zb
a
f(x)dx ≤(b−a)f(a+b
2).
2. Find the limit of the sequence
an=1
n r1 + 1
n+r1 + 2
n+. . . +r1 + n
n!.
3. Find Z∞
0
1
(1 + x2)(1 + xα)dx.
4. Find Zπ
0
ln sin x dx.
5. Find Zπ
0
xsin x
1 + cos2xdx.
6. Show that if fis continuously differentiable on [0, a] and f(0) = 0, then
Za
0
f(x)dx =Za
0
(a−x)f0(x)dx.
7. Show that for any nnatural number
ln n
n+ 1 <1
pn(n+ 1).
8. Find
Zπ/2
0
1
1 + (tan x)√2dx.