PROBLEM SET VIII
“FUNDAMENTALS”
DUE FRIDAY,  NOVEMBER 
In this problem set, we will use define some important functions using integrals. Use only the definitions given here in the following exercises.
Definition. Define a function arctan : R . R by the following formula:
∫ x
1
dt.
arctan(x) =
2
0 1+t
Exercise . Show that arctan is increasing and that for any x ∈ R, one has
arctan(−x) = − arctan(x)
and
| arctan(x)| ≤ |x|.
Definition. Now define π as the real number 4 arctan(1).
Exercise . Show that the image arctan(R) is the open interval (−π/2, π/2).
[Hint: show that for any ε > 0, there exists a real number N > 0 such that for
any x > N, one has π/2 − ε < arctan(x) < π/2.]
Exercise . Prove that arctan : R .
(−π/2, π/2) is a bijection of class C∞ .
Definition. Denote by R+ the ray (0, +∞). Define a function log : R+ .
the formula
∫ x
1
log(x) =
dt.
1 t
R by
Exercise . Prove that for any x, y ∈ R+ , one has
log(xy) = log(x) + log(y).
Exercise . Show that log is increasing, and show that the image log(R+ ) is R.
Exercise . Prove that log : R+ .
R is a bijection of class C∞ .
Exercise . Prove that the inverse exp : R . R+ of the function log is a differentiable function such that for any elements x, y ∈ R,
exp′ (x) = exp(x),
exp(0) = 1,
and

exp(x + y) = exp(x) exp(y).