Bowdoin Math 1750B, Fall 2015 – Homework #2 (revised)
More on integrals (definite, indefinite, improper), properties of even and odd functions
Due on Thursday, September 17 (at the beginning of class)
This homework assignment is on two pages.
Read sections 6.2, 6.4, 7.1, 7.2, 7.3, 7.4, 7.6 of Calculus by Hughes-Hallett et al., 6th edition. Most of the
material should be rather familiar, possibly with the exception of 6.4 (on the second fundamental theorem of
calculus) and 7.6 (on improper integrals); we will cover these two sections on Tuesday, September 15.
Solve the following problems. To obtain full credit, please write clearly and show your full reasoning.
Please solve the problems in the order given and staple your homework together. Also, write your name
and Bowdoin ID# on the top of the first page.
Problem 2A. Evaluate the following integrals (some definite, others indefinite):
Z
sin(ax + b) dx, where a and b are constants with a 6= 0.
(a)
Z
cos2 x sin3 x dx
Z
x2 sin x dx
(b)
(c)
Z
9
(d)
4
√
Z
ln y
√ dy
y
3
(e)
1
Z
(f )
√
1
arctan
dx
x
1
dx
1 − x2
Z
1
dx
1 + x2
Z
p
(h)
x3 1 − x2 dx
(g)
x2
√
Problem 2B. Evaluate the following improper integrals:
Z ∞
1
(a)
dx
x(ln
x)2
3
Z 1
1
√
dx
(b)
3
x
0
Z π
√
1
√ e− x dx
(c)
x
0
Z π/2
sin x
√
(d)
dx
cos x
π/4
1
Problem 2C. Compute the area A of the region in the xy-plane that is simultaneously bound by the
lines y = 1 and y = e, and by the curves y = x2 and t = ex (see Figure 1 below). Extra points if you
compute it using two methods: (i) with integration with respect to x, and (ii) with integration with respect
to y (to do this, you have to first invert the functions y = f (x) = x2 and y = g(x) = ex , by solving for x for
each one of them). Of course, at the end you should get the same result!
{
I
I
x-
I
I
~"'6
(
,
(
I
!
e
- .- -
,
-
I
-I-
./
,
."
,
....... -
I
I
i
,"
I
I
o
-
Ie
t
x
Figure 1. The region for Problem 2B, whose area you have to compute.
Problem 2D. Assume that the function f : R → R has derivative function f 0 , and show that:
(a) If f is even, then f 0 is odd.
(b) On the other hand, if f is odd, then f 0 is even.
(c) To visualize the above general fact with an example, sketch the graph of f (x) = arctan(x), x ∈ R, as
well as the graph of its derivative, on the same plot.
Hint: for parts (a) and (b), you must show that the statements are true for any function f . So for (say)
part (a) you should first write the definition of even function; then apply the chain rule. . .
Problem 2E (odd and even part of a function). Consider a function f : R → R.
(a) Show rigorously that if f is odd, then f (0) = 0 (start from the definition of “odd function”).
(b) Show that if f is simultaneously even and odd, then it must be the zero function.
(c) Suppose that f is generic (not
necessarily even or odd). The new functions fe (x) = 21 f (x) + f (−x)
and fo (x) = 12 f (x) − f (−x) , defined for x ∈ R, are called, respectively, the even part and the odd part
of the function f . Show that: (i) fe is even; (ii) fo is odd; (iii) f (x) = fe (x) + fo (x), for all x ∈ R.
(d) Now, consider the function f (x) = ex , and compute its even and odd parts. Do you recognize these as
known functions?
Hint: remember that by the “zero function”, which we indicate with f = 0, we intend the function that is
zero everywhere, i.e. such that f (x) = 0 for all the x’s in its domain of definition.
2