Integral Formulas
Z
1.
Z
2.
Z
3.
un du =
1
un+1 + c if n 6= −1.
n+1
du
= ln |u| + c
u
eu du = eu + c
Z
Z
sin u du = − cos u + c
4.
Z
5.
cos u du = sin u + c
1
sin u du = [u − sin u cos u] + c
2
2
Z
1
cos2 u du = [u + sin u cos u] + c
2
Z
sec u du = ln | sec u + tan u| + c
6.
Z
csc u du = ln | csc u − cot u| + c
7.
Z
Z
tan u du = ln | sec u| + c
8.
cot u du = ln | sin u| + c
Z
1
n−1
n−1
sin u du = − sin
sinn−2 u du + c
9.
u cos u +
n
n
Z
Z
n−1
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
n
n−1
12.
tan u du =
tan
u − tann−2 u du + c, if n > 1.
n−1
Z
Z
1
n−2
n
n−2
cscn−2 u du + c, if n > 1.
13.
csc u du = −
csc
u cot u +
n−1
n−1
Z
Z
1
n
n−1
14.
cot u du = −
cot
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
n
Trigonometry Formulas
sin(u + v) = sin u cos v + cos u sin v
cos(u + v) = cos u cos v − sin u sin v
tan(u + v) =
tan u + tan v
1 − tan u tan v
sin(u − v) = sin u cos v − cos u sin v
cos(u − v) = cos u cos v + sin u sin v
tan(u − v) =
tan u − tan v
1 + tan u tan v
1
sin2 u = (1 − cos 2u)
2
1
cos2 u = (1 + cos 2u)
2
sin 2u = 2 sin u cos u
cos 2u = cos2 u − sin2 u
= 1 − 2 sin2 u
= 2 cos2 u − 1
2 tan u
tan 2u =
1 − tan2 u
sin a + sin b = 2 sin
a−b
a+b
cos
2
2
sin a − sin b = 2 cos
a+b
a−b
sin
2
2
cos a + cos b = 2 cos
a+b
a−b
cos
2
2
cos a − cos b = −2 sin
a+b
a−b
sin
2
2
Calculus II,
NAME
Tuesday, September 18, 2007
Rec. Instr.
Exam 1
Rec. Hour
In each of the following problems, show your work, and organize your work clearly, so
as to maximize the opportunities for “partial credit”. For example, many integration
problems have several involved steps, and you should demonstrate that you know what
the steps are, and how to proceed with them.
(12 pts) 1. Evaluate the limits:
(a) lim
x→∞
(b)
100x
x2 − 100x
lim+ (1 − x)1/x
x→0
(15 pts) 2. Find a function f (x) whose derivative is xe−x .
Page 2 of 5
x3
√
(20 pts) 3. Evaluate:
dx
x2 + 4
(There are a lot of steps, and a lot of opportunities for partial credit – so try
to present the steps in an organized fashion.)
Z
Page 3 of 5
Z
(12 pts) 4.
Evaluate:
0
Z
(6 pts) 5.
Is
2
answer.)
3
1
dx
(x − 2)(x − 3)
dx
(x − 2)(x − 3)
convergent? Divergent? (Justify your
Page 4 of 5
(15 pts) 6. Find the volume of revolution obtained from the region bounded by
y = cos x and the lines x = 0, x = π/2, y = 0, revolved about the x-axis.
(10) 7. Same problem as 6, but revolve about the line y = 1.
Page 5 of 5
Z
(10 pts) 8. Evaluate
x dx
√
.
x−4