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University University of Winnipeg
Course MATH-1104-003 Introduction to Calculus II
Year 2026
Author Christine
Subject Mathematics
File Type PDF

19 views 15 pages

Assignment Instructions

MATH-1104 — Introduction to Calculus II University of Winnipeg | Winter 2026 | Dr. Ross Stokke
Test 1 — Study Guide & Practice Problems
Preparation material for Test 1 | Saturday, February 7, 2026
Key Topics
• Antiderivatives and indefinite integrals
• Differential equations
• Definite integral as a limit of Riemann sums
• Integration properties
• FTC I and FTC II
• Substitution rule
• Symmetry and the definite integral
Proof List
You should know the proofs of the following theorems for the test:
bbb
1. If f and g are continuous functions, then ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx.
aaa
2. The Mean Value Theorem for Integrals (Note: the proof in the textbook is not acceptable, since
it pre-assumes FTC.)
3. The Fundamental Theorem of Calculus, Part I
4. The Fundamental Theorem of Calculus, Part II
Textbook Problems to Work Through
• Section 4.9: Problems 29–53 (odd)
• Section 5.2: Problems 19, 20, 25, 26, 51–59
• Section 5.3: Problems 3, 4, 9–19 (odd), 67–71

Extra Integration Problems
Problem 1
For each of the following, express the integral as a limit of Riemann sums. Do not evaluate the
limit.
32
(a) ∫ (4x − 2) dx
0
(b) ∫ −2
(e) w(x) = ∫ 4 Problem 4
22 √(5 − x ) dx
53
(c) ∫ x sin(x ) dx
1
Problem 2
For each of the following, express the limit as a definite integral ∫
(3/n) √(2 + 3i/n) ln(1 + 4i/n) · (4/n)
n n→∞ i=1
(a) lim Σ (b) lim Σ
n
n→∞ i=1
n4
(c) lim Σ [1 / (1 + (2 + 2i/n) )] · (2/n)
Problem 3
n→∞ i=1
Find the derivatives of the following functions.
x3
(a) f(x) = ∫ √(1 + t ) dt
0
(b) f(x) = ∫ x
sin x 2 (t
(d) h(x) = ∫
sin x
(c) g(x) = ∫ 1
+ 2t) dt
− √t) dt sin(√t + 1) dt
243
(a) ∫ (2x − 8x ) dx
1
(b) ∫ 1/√(1 − x −1/2
9
(c) ∫ 1/√x dx
4
(d) ∫ 8
27 4/3
x dx
π/4 2 (e) ∫ sec
x dx
0 322
(f) ∫
(g)∫ x dx
1
1
4 −1 (h)∫ x dx
π/2
(i) ∫ sin x dx
−1
−π
2 1/3
t / (1 + t) dt
2 x −1
1/2 2
x3 (t
Use FTC II to evaluate the integral or explain why this is not possible.
) dx
(3x − 6/x ) dx 3 −2
b a
f(x) dx.

π2
(j) ∫ tan x dx
(k) ∫ −2
0
22
(3x + x − 5) dx

Problem 5
Evaluate each of the following definite integrals.
3 0
3
0
23
(c) ∫ f(x) dx where f(x) = { x if 0 ≤ x < 1, x if 1 ≤ x ≤ 2 }
0
Problem 6
Evaluate the integrals.
(a) ∫ (b) ∫
|2x − 1| dx
(2 + 3|1 − x|) dx
(a) ∫ 2/(5 − 6x) dx 2
(b) ∫ sec x / √(1 + tan x) dx 27
(c) ∫ (1/x )(1 − 1/x) dx 2
(d) ∫ π/(64 + x ) dx 22
(e) ∫ x sec(x ) tan(x ) dx 3
(f) ∫ cos x / (4 + sin x) dx (g) ∫ sin(√x) / √x dx
2/3 (h) ∫ (7x − 3) dx
(i) ∫ (cos(3x) − sin(3x)) dx (j) ∫ x / √(2x + 1) dx
2
(k) ∫ 1/(9 + 4x ) dx
457
(l) ∫ x / (x − 1) dx
5
(m) ∫ sin (3x) cos(3x) dx
223
(n) ∫ (2x + 3x ) sin(x + x ) dx
(o) ∫ cos x · sin(sin x) dx 2
(p) ∫ x / √(1 − x) dx
Problem 7
Evaluate the integrals.
2
(a) ∫ 3/(x√(x − 9)) dx
x2
(b) ∫ (6 + 5/√(25 − x ) + 8/(2x + 7)) dx
22
(c) ∫ cot x · csc x dx
(d) ∫ (4x − 3)/(4x 2
23
− 6x + 9) dx
(e) ∫ 1/cos (3x) dx (f) ∫ ■√(5 − 3x) dx

22
(g) ∫ 1/√(4 − x ) dx
323
(h) ∫ x /√(x + 9) dx
0
2
(i) ∫ 3/(5 + x ) dx
(j) ∫ x
/√(5 − x ) dx −1/2
√2
8 0
53
(k) ∫
(l) ∫ tan x · sec x dx
(1 + x) dx
54
32
(m) ∫ x √(x − 3) dx
x 2x
(n) ∫ e /(1 + e ) dx
2
(o) ∫ (sec x + sec x tan x)/(sec x + tan x) dx
−1 2 (p) ∫ tan (5x)/(25x
4
(q) ∫ x/(11 + x ) dx
(r) ∫ (s) ∫
π/2 22
(t) ∫ −11
(1 − cos 113 2
+ 1) dx
x) sin x dx
0
11
11 3
x /(3 + cos x) dx
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