## Transcribed Text

1. Let f : [−a, a] → R be a continuous function where a > 0. If f satises that
Z a
−a
f(x)g(x)dx = 0
for every integrable even function g : [−a, a] → R, show that f is an odd function.
2. Let f : [0, 1] → R be a dierentiable function with f(0) = 0 and f
0
(x) ∈ (0, 1), x ∈
(0, 1). Show that
Z 1
0
f(x)dx!2
>
Z 1
0
(f(x))3
dx
(Hint: Show that the function F(x) = (R x
0
f(t)dt)
2 −
R x
0
(f(t))3dt is increasing.)
3. For what values of k the improper integral
Z ∞
2
dx
x
k
ln x
converges?
4. Apply Bolzano-Cauchy Criterion to show that the improper integral
Z ∞
0
sin x
x
dx
converges.
5. Let f : [a, b] → R be an integral function. Dene
f+(x) = (
f(x) if f(x) ≥ 0,
0 if f(x) < 0,
and f− =
(
0 if f(x) ≥ 0,
−f(x) if f(x) < 0.
Prove that f+, f− : [a, b] → R are integrable and
Z b
a
f(x)dx =
Z b
a
f+(x)dx −
Z b
a
f−(x)dx.
6. Let f : [a, b] → R be a Riemann integrable function such that
∃α > 0 3 ∀x ∈ [a, b], f(x) ≥ α.
Show that the function g(x) = 1
f(x)
is also Riemann integrable on [a, b].
7. Compute the following indenite integral
Z
dx
1 + x
4
18. Let f : (0, 1) → R be a continuous function such that the improper integral
Z π
0
f(sin x)dx
converges. Show that
Z π
0
xf(sin x)dx = π
Z π
2
0
f(sin x)dx
1. Solve the following problems
a) Let f : [a,∞) → R be a function such that the improper integral
Z ∞
a
f(x)dx
converges. Assume the existence of the (nite) limit
limx→∞
f(x) = α
Show that α = 0.
b) Assume that f : [a,∞) → R be a continuously dierentiable function such that the
improper integrals
Z ∞
a
f(x)dx
and
Z ∞
a
f
0
(x)dx
converge. Show that
limx→∞
f(x) = 0.
2. Compute the following improper integrals
Z ∞
0
xe−αx cos(βx)dx, Z ∞
0
xe−αx sin(βx)dx.
3. Solve the following problems
a) Compute improper integral:
Z π
2
0
x cot xdx
(Hint: Use the integration by parts).
b) Compute improper integral:
Z 1
0
arcsin x
x
dx.
(Hint: Use the integration by parts).

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