Exercises 35

(15) Show that

x

x0

f(u) du = O

x

x0

g(u) du

if f(x) = O(g(x)) and f and g are integrable on bounded intervals.

(16) Prove that if f and g are real-valued continuous functions on a closed

and bounded interval [a, b] and g is positive there, then

b

a

f(x)g(x) dx = f(c)

b

a

g(x) dx

for some c ∈ (a, b). This is called the first mean value theorem for

integrals.

(17) a) Prove that if f and g are real-valued continuous functions on a closed

and bounded interval [a, b] and f is monotone there, then

b

a

f(x)g(x) dx = f(a)

c

a

g(x) dx + f(b)

b

c

g(x) dx

for some c ∈ (a, b). This is called the second mean value theorem for

integrals. The second mean value theorem is useful for the estimation

of integrals with an oscillatory factor in the integrand.

b) Show that

2π

0

f(x) cos(x) dx ≤ |f(0) − f(2π)|

if f is a continuous monotone function.

(18) Show that

lim

h→0

b

a

f(x + h) − f(x)

h

dx = f(b) − f(a)

if f is a continuous function on an open interval containing the closed

interval [a, b]. Establish an integration by parts formula for two contin-

uous functions, one of which is monotone.

(19) Make a guess for the asymptotic behavior of the sum

p≤x

1

√

p

and use partial summation and estimates of Chebyshev to show that

your guess has the right order of growth.

(20) Show that

pq≤x

log(p) log(q) x log(x)

where p and q denote primes.