Integral Formulas
1.
Z
un du =
2.
Z
du
= ln |u| + c
u
3.
Z
eu du = eu + c
4.
Z
sin u du = − cos u + c
5.
Z
1
sin2 u du = [u − sin u cos u] + c
2
6.
Z
sec u du = ln | sec u + tan u| + c
7.
Z
csc u du = ln | csc u − cot u| + c
8.
Z
1
un+1 + c if n 6= −1 .
n+1
tan u du = ln | sec u| + c
Z
cos u du = sin u + c
Z
Z
1
cos2 u du = [u + sin u cos u] + c
2
cot u du = ln | sin u| + c
Z
1
n
−
1
9.
sinn u du = − sinn−1 u cos u +
sinn−2 u du + c
n
n
Z
Z
1
n−1
n
n−1
10. cos u du = cos
u sin u +
cosn−2 u du + c
n
n
Z
Z
1
n−2
n
n−2
11.
sec u du =
sec
u tan u +
secn−2 u du + c , if n > 1 .
n−1
n−1
Z
Z
1
12.
tann u du =
tann−1 u − tann−2 u du + c , if n > 1 .
n−1
Z
Z
1
n
−
2
13.
cscn u du = −
cscn−2 u cot u +
cscn−2 u du + c , if n > 1 .
n−1
n−1
Z
Z
1
14.
cotn u du = −
cotn−1 u − cotn−2 u du + c , if n > 1 .
n−1
Z
Z
15.
sinh u du = cosh u + c ,
cosh u du = sinh u + c .
Z
Calc II, Final Exam
Name
May 15, 2002
Rec. Instr.
Rec. Hour
page
1
2
3
4
5
6
7
8
total
points
Work as many of the following problems as you can in one hour. To
assure partial credit, show your work. No calculators are permitted for
this exam.
(10) 1. Carbon extracted from a tooth found at an archaeological dig
contains 15 as much radioactive as is found in teeth of living creatures.
The half-life of 14C is 5700 years. Even though you don’t have a
calculator, find an expression for the age of the tooth.
1
(10) 2. Find the derivative of the function y = xsin x at the point x =
y = π/2 .
(18) 3. Evaluate the following limits. (Answers may be: No limit exists,
the limit is ∞ , or −∞ , or some specific real number which it is your
task to determine.)
(a) lim
x→0
x
tan x
(b) lim xsin x
x→0+
et
(c) lim 100
t→∞ t
−t
2
4. Find the following antiderivatives.
Z
(11) (a)
x2e−xdx
(11) (b)
Z
sin x tan x dx
(11) (c)
Z
dx
(x − 1)(x + 1)2
3
(11) (d)
Z √
(10) (e)
Z
1
∞
4x2
dx
xP
+ 9 dx
Hint:
Z
sec3 θdθ =
1
1
sec θ tan θ + ln | sec θ + tan θ| + c
2
2
( p constant, p > 1 ) [SUGGESTION: Try p = 3 .]
(12) 5. Determine whether the given sequence {an}∞
n=0 converges, and
find its limit if it does converge.
√
1 + (−1)n n
(a) an =
n+1
(b) an = 1 +
1 1
1
+ + ··· + n
3 9
3
4
(18) 6.(a) Write down the MacLaurin series for xe−2x , using what you
already know about the MacLaurin series for ex .
(b) What is the radius of convergence of the series in part (a)?
(c) Write down the 3rd Taylor polynomial P3(x) , for xe−2x , at a = 0 .
5
(24) 7. Find the interval of convergence for the following two power series.
(Justify your treatment of endpoints.)
(a)
∞
X
2n n
x
ln(n)
∞
X
n
n=5
(x − 1)n
(b)
(−1)
n
2
n=0
6
(18) 8(a) Find a power series expression for the function f (x) =
√
16 + x4 .
(b) What is the radius of convergence for the series in part (a)?
√
(c) Find an antiderivative of 16 + x4 , using part (a). Write out the
first four terms of this (antiderivative) series.
dy
d2y
t
(12) 9. Find
and
at
x
=
1
,
for
the
parametric
curve
x
=
e
,
dx
dx2
y = sin t .
7
(12) 10. Find the area of the surface obtained by revolving the curve
1
x = t , y = t2 (0 ≤ t ≤ 4) around the y-axis .
2
(12) 11. Find the volume of the solid obtained by revolving the curve
x = t2 , y = sin t 0 ≤ t ≤ π2 around the x-axis .
8