Download Quiz 4 Information

Quiz 4 Information
Math 208
page 1
Quiz 4 will cover the following material: (all handouts posted on the web site so far)
1. All material for Quizzes 1-3
2. Di¤erentiate any function, including logarithmic, exponential, and inverse trigonometric functions.
3. Apply the fundamental theorem to compute de…nite integrals and di¤erentiate functions de…ned using de…nite
integrals.
4. Graph trigonometric functions (all 12 of them) and state their basic propoerties, di¤erentiate them and
integrate them. (Exception: we did not yet integrate sec 1 x and csc 1 x; so you don’t need to know those.)
5. Integrate using substitution, trigonometric substitution, integration by parts, and partial fractions with linear
factors. Integrate trigonometric and inverse trigonometric functions.
6. Approximate logarithmic expressions using the de…nition of ln x as
Zx
1
dt.
t
1
Sample Quiz 4
dy
1. Compute
if y =
dx
sec
Zx
1
p
4
1
dt
1 + e2t
2. Compute each of the following integrals.
a)
b)
c)
d)
e)
f)
g)
h)
Z
Z
Z
Z
Z
Z
Z
Z
1
(x + 1) (x
dx
1)
x
(x + 1) (x
1)
dx
i)
Z
p
j)
Z2
p
x x
1)
dx
Z3
k)
5
dx
2
x +x
sin 2x
dx
1 + cos 2x
l)
m)
Z
n)
Z
o)
Z2
sec3 xdx
dx
Z
0
p)
Z
p
1 + cos xdx
0
1dx
x2
q)
Z=4
r)
Z
1
dx
arctan x
dx
1 + x2
s)
x2 cos (2x) dx
t)
p
p
1
9x2
1
4
tan2 xdx
0
1
2
1
dx
2
x + 2x + 2
9
9
dx
1
x2
(x + 1) (x
1
p
4x2
x
4x2
x2
25
dx
dx
u)
v)
Z
Z
Z
Z
sec3 x tan3 xdx
ex
dx
ex + 1
xe
x dx
sin 2x sin 10x dx
x4 ln x dx
Last revised: February 9, 2017
Quiz 4 Information
Math 208
page 2
Answers
sec x tan x
1. p
4
1 + e2 sec x
2. a)
1
ln jx
2
d) 5 ln jxj
g)
j)
n)
1j
1
ln jx + 1j + C
2
5 ln jx + 1j + C
b)
1
ln x2
2
e) arctan jx + 1j + C
1
(sec x tan x + ln jsec x + tan xj) + C
2
16
15
k)
1
1
ln 2 + ln 3
2
2
p
1
ln 18x + 6 9x2
3
s) ln (ex + 1) + C
e
x
o)
xe
h)
x
2
+C
c) x +
p
1
ln 2x + 4x2
2
m)
p
p) 2 2
u)
1
sin 8x
16
1
ln jx
2
1
ln jx + 1j + C
2
1j
1
ln jcos 2x + 1j + C
2
f)
1
arctan2 x + C
2
l)
25 + C
t)
1 +C
9 +C
1
x cos 2x
2
q) 1
i)
1p 2
4x
4
9+C
1
1
sin 2x + x2 sin 2x + C
4
2
r)
4
1
sin 12x + C
24
1
sec5 x
5
v)
1
sec3 x + C
3
1 5
x ln x
5
1 5
x +C
25
Last revised: February 9, 2017