MATH1014
Calculus II
L01 (Spring 2025)
Problem Set 3
1. (a) Let m and n be non-negative integers. Evaluate the following integrals, distinguishing all possible cases for m and n.
(b) Let n be apositive integer and let f: ℝ → ℝ be afunction defined by
f(x) = a1 sin x + a2 sin 2x + ⋯ + an sinnx ,
where a1, a2, … , an are real numbers. Show that we must have
2. Evaluate the following antiderivatives.
Hint: In (d),first consider ∫ ex 2 dx.
3. Evaluate the limit
Hint: Take natural logarithm.
4. Let a > 0 and let f: [−a, a] → ℝ bean odd continuous function. Show that
5. The following are “ proofs” of some obviously false statements. Point out what is wrong in each of these “ proofs”.
(a) A “ proof” of the statement that “π = 0”.
(b) A “ proof” of the statement that “every integral equals zero” :
(c) A “ proof” of the statement that “0 = 1”.
6. Let f be afunction which is continuously differentiable on [0, 1].
(a) For every a, b ∈ [0, 1], show that
(b) Let n ≥ k ≥ 1 be integers. Using the result from (a) and the generalized Mean Value
Theorem for integrals (Example 5.49 (a)), show that there exists such that
(c) Now for each n ∈ ℕ, we let Show that
Hence using the result from (b),deduce that
7. Let f: [0, +∞) → ℝ be the function defined by f(x) = xex. (a) Show that f is strictly increasing.
(b) Now f is one-to-one according to (a), so we let g be the inverse of f, i.e. g = f −1 .
(i) Write down the domain of g. Show that
for every x in the interior of the domain of g.
(ii) Using the result from (b) (i) or otherwise, evaluate the antiderivative ∫ g(x)dx,
expressing your answer in terms of g and other elementary functions only.
(iii) Hence,or otherwise, evaluate the integral
8. (a) Let n beanon-negative integer, and let f: ℝ → ℝ be the polynomial f(x) = (x2 − 1)n.
(i) Show that (x2 − 1)f′ (x) − 2nxf(x) = 0 for every x ∈ ℝ .
(ii) Hence, show that
(x2 − 1)f(n+2)(x) + 2xf(n+1)(x) − n(n + 1)f(n)(x) = 0
for every x ∈ ℝ .
Hint: Recall “ Leibniz rule” in chapter 3. Part (a) is almost the same as Example 3.69. (b) For each non-negative integer n, let pn : ℝ → ℝ be the function
(i) Using the result from (a) (ii), show that
for every non-negative integer n.
(ii) Hence deduce that if m and n are distinct non-negative integers, then
9. For each non-negative integer n, let
(a) For each positive integer n, show that
Hence show that
(b) Using the result from (a), find the value of In in terms of n.