MTH 200 — Midterm 2 Review
Time Limit: 90 Minutes
Winter 2025
Instructor: Ricky Ng
About Midterm 2
• This is an closed book exam. The only electronic device allowed for computation is a scientific, non-graphing calculator.
• The time limit is 90 minutes. It has 2 sections and contains 9 questions. Midterm 2
covers the following sections in the textbook:
§7.1 — §7.4, §8.3
• Part 1 consists of 5 multiple-choice questions with a total of 10 points. You do not
need to provide any explanation to justify your answer.
• Part 2 consists of 4 free response questions with a total of 40 points. Your solution
will be graded based on calculation, presentation, and reasoning. Answers without
proper work shown will not be given any credit.
MTH 200
Midterm 2 Review - Page 2 of 5
Sample Questions
Integration By Parts:
Z
Z
u dv = uv −
General Antiderivatives:
Z
1
dx = ln |x| + C
x
Z
xp+1
xp dx =
+ C, p ̸= −1
p+1
Z
cos(x) dx = sin(x) + C
Z
tan(x) dx = − ln | cos(x)| + C
v du
Z
Z
ex dx = ex + C
x
1
1
arctan
+C
dx
=
x 2 + a2
a
a
Z
sin(x) dx = − cos(x) + C
Z
cot(x) dx = ln | sin(x)| + C
Special Trig. Antiderivatives:
Z
Z
sec θ tan θ dθ = sec θ + C
sec θ dθ = ln | sec θ + tan θ| + C
Z
Z
1
3
sec θ dθ = (sec θ tan θ + ln | sec θ + tan θ|) + C
sec2 θ dθ = tan θ + C
2
Z
Z
csc θ cot θ dθ = − csc θ + C
csc θ dθ = − ln | csc θ + cot θ| + C
Z
Z
1
3
csc θ dθ = − (csc θ cot θ + ln | csc θ + cot θ|) + C
csc2 θ dθ = − cot θ + C
2
Pythagorean Identities:
2
Sum and Difference:
2
cos θ + sin θ = 1
1 + tan2 θ = sec2 θ
cot2 θ + 1 = csc2 θ
Half-Angle Formulas:
1 − cos(2θ)
sin2 θ =
2
1
+
cos(2θ)
cos2 θ =
2
sin(A ± B) = sin A cos B ± sin B cos A
cos(A ± B) = cos A cos B ∓ sin A sin B
Double-Angle Formulas:
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos2 θ − sin2 θ
= 1 − 2 sin2 θ = 2 cos2 θ − 1
Product Identities:
1
[cos(A − B) − cos(A + B)]
2
1
sin(A) cos(B) = [sin(A − B) + sin(A + B)]
2
1
cos(A) cos(B) = [cos(A − B) + cos(A + B)]
2
sin(A) sin(B) =
MTH 200
Midterm 2 Review - Page 3 of 5
Sample Questions
Center of Mass and Moments
• n point-mass system:
My =
n
X
mi xi ,
Mx =
i=1
x̄ =
n
X
mi y i
i=1
My
,
m
ȳ =
Mx
m
• Lamina with region R with uniform density ρ: For A the area of R,
Z b
Z
ρ b
xf (x) dx,
Mx =
My = ρ
[f (x)]2 dx
2
a
a
Z
Z
1 b
1 b1
x̄ =
xf (x) dx,
ȳ =
[f (x)]2 dx
A a
A a 2
• Lamina with uniform density ρ and region R between two curves f (x) ≥ g(x): For A
the area of R,
Z
Z
1 b1
1 b
2
2
x f (x) − g(x) dx,
ȳ =
x̄ =
[f (x)] − [g(x)] dx
A a
A a 2
MTH 200
Midterm 2 Review - Page 4 of 5
Sample Questions
Sample Questions
These sample questions are for you to have an idea on the types of questions and the level
of difficulty in the exam. If you feel comfortable with the questions, you should be fine with
the midterm. ANSWERS will not be provided.
1. Integration with Sine and Cosine Functions. Evaluate the following. Clearly show
your work.
Z
Z
2
(d)
cos(−2θ) sin(3θ) dθ.
(a)
sin (3θ) cos(3θ) dθ.
Z
Z
4
2
(e)
sin(−3θ) sin(5θ) dθ.
(b)
sin (θ) cos θ dθ.
Z
Z
3
99
(f)
cos(θ) cos(4θ) dθ.
(c)
cos (θ) sin (θ) dθ.
2. IBP. Evaluate the following indefinite integrals. Clearly show your work.
Z
Z
2
(a)
x sin(x) dx.
(d)
x2 ln(x2 ) dx.
Z
Z
2 2x
(b)
x e dx.
(e)
arctan(x) dx.
Z
Z
3
2
(c)
x cos(x ) dx.
(f)
ln(x2 + 1) dx.
3. Tabular Method. Evaluate the following indefinite integrals by the tabular method.
Z
Z
4
(a)
x sin(2x) dx.
(c)
e−2x cos(3x − π) dx.
Z
Z
3
2
x
(b) (x − x + 1)e dx.
(d)
ex+2 sin(5x) dx.
4. Trig Sub. For each of the following problems:
(i). Make an appropriate right triangle for the trigonometric substitution.
(ii). Set up the trigonometric integral.
(iii). Evaluate the integral. You may need to use the formula sheet provided.
Z
Z
x2
x2
√
√
(a)
dx.
(d)
dx.
9 − x2
x2 + 1
Z
Z √
1
36 − x2
√
(e)
dx.
(b)
dx.
2
x 1 − x2
x
Z
Z √
3
√
(f)
dx.
(c)
x2 + 25 dx.
4x2 − 9
MTH 200
Midterm 2 Review - Page 5 of 5
Sample Questions
5. PFD. Give the form of the partial fraction decomposition for the following rational
functions. If necessary, apply the long division first. Do not evaluate.
x
.
(x − 2)(x + 3)
x3
.
(b)
(x + 1)(x + 2)
x2 + 2x + 5
(c)
.
(x + 1)2 (2x − 5)
1
(d)
.
(x − 5)3 (3x2 + 5)
(a)
x3
.
(2x2 + 3)(x2 + 4)
x
.
(f) 2
x + 3x − 8
x3 + 1
(g)
.
(x2 + 4)(x − 3)x2
(e)
6. PFD. For each of the following problems:
(i). Write down the form of partial fraction decomposition.
(ii). Find the partial fraction decomposition. Clearly show your algebraic work.
(iii). Evaluate the indefinite integral.
Z
5x
dx.
(a)
(x + 2)(x − 3)
Z
3
(b)
dx.
(2x − 5)(x + 1)
Z
9x
dx.
(c)
(x + 1)2 (x − 2)
Z
x2 + 2x − 1
dx.
(d)
(x2 + 1)(x − 1)
Z
x2
(e)
dx.
x2 + 2x − 3
Z 3
x +3
(f)
dx
x2 − 1
7. Centroid and Moments. For each of the following regions with uniform density ρ:
(i). Find its center of mass (x̄, ȳ)
(ii). Calculate its moments Mx and My
(a) m1 = 4 at (5, −1), m2 = 2 at (−3, −9), m3 = 5 at (1, 1)
√
(b) The region R bounded by the x-axis and y = x, where 0 ≤ x ≤ 4, with uniform
density ρ = 5.
(c) The region R bounded by the x-axis and y = ln(x), where 1 ≤ x ≤ e, with uniform
density ρ = 2.