Midterm 2 Practice
Exam Date: October 31, 2014
You will have an hour and 20 minutes to complete midterm 2. On this exam you can bring
one 3”x 5” notecard with any information you find useful for the exam. Please note: Nothing
bigger will be allowed - it is one 3” x 5” notecard!
1. Using integration by parts, integrate the following:
Z
(a)
ln2 z dz
Z
(b)
5
√
2x ln x3 dx
1
Z
(c)
Z
(d)
eat sin t dt
xex dx
1
2. Evaluate each integral.
Z
(a)
cos6 θ sin2 θ dθ
Z
(b)
sin 4y cos 5y dy
Z
(c)
Z
(d)
tan−3 x sec4 x dx
dt
+ 1)
t (t1/6
2
3. Use partial fractions to integrate the following:
Z
2x2 − x − 20
(a)
dx
x2 + x − 6
Z
(b)
x4
1
dx
− 16
4. Solve y 0 = y(1 − y), if y(0) = 0.5.
3
5. Evaluate the following limits.
2x − sin x
x→0
x
(a) lim
(b)
lim
x→(π/2)−
(cos 2x)x−π/2
1
(c) lim csc x −
x→0
x
(d) lim+
x→0
ln sin2 x
3 ln tan x
4
6. The following is a probability density function

5e−5x if x ≥ 0
f (x) =
0
otherwise.
Z
∞
f (x)dx = 1).
(a) Show that f(x) is a pdf (i.e.
−∞
(b) Find the mean, µ and the variance, σ 2 . Where,
Z ∞
xf (x)dx
µ=
−∞
2
Z
∞
(x − µ)2 f (x)dx
σ =
−∞
5
7. Integrate the following:
Z 3
x
√
(a)
9 − x2
0
8. Use the comparison test to show the following converges or diverges:
Z ∞
dx
√
(a)
dx
x6 + x
1
Z ∞
ln x
dx
(b)
x
3
6