MATH1014
Calculus II
Problem Set 4
L01 (Spring 2025)
1. For each of the following rational functions f, evaluate the antiderivative ∫ f(x)dx.
2. Evaluate the following antiderivatives.
3. (a) Using the factorization x4 + 1 = (x2 − √2x + 1)(x2 + √2x + 1), evaluate
(b) Using (a) and the substitution u = √tan x, evaluate
4. Evaluate the antiderivative
using the substitution
5. (a) Show that the polynomial x 3 + 3x + 1 has exactly one real root.
(b) Let r be the real root of x 3 + 3x + 1. Using a partial fraction decomposition, evaluate
in terms of r.
6. Evaluate the antiderivatives of each of the following trigonometric rational functions f.
7. Let a be apositive real number. Evaluate
for each of the following cases:
(a) 0 < a < 1,
(Try to find antiderivatives just on (−π, π); antiderivativeson ℝ are too complicated.)
(b) a = 1,
(c) a > 1.
8. Evaluate the antiderivative
using the substitution t = tan(x/2).
9. Evaluate each of the following improper integrals if it converges.
Hint: In (c), break the interval into two halves, and let in the second half.
10. Let f: [0, +∞) → ℝ be the function
(a) Find f ′ (x) for every x ∈ (0, +∞).
(b) Evaluate the improper integral