Math 11B
Spring 2008
Review problems
Copyright by Hongyun Wang, UCSC
Review problems for the final exam
(1)
Find
e
(2)
Find
sin ( 2x + 5 ) dx
(3)
Find
3x
(4)
Find the general solution of
(5)
Find the solution of the initial value problem
3x+2
2
dx
+
4 5
+ dx
x x2 t
dy
=
dt t + 2
)
(6)
d
sin u 2 + 2 du
Find
dx a
(7)
d
Find
dx
(8)
d
Find
sin
dx x2+1
(9)
Find
x2 + 3
a
sin
ex + x
1
x +1
y ( 0) = 2
with
(
x
dy
=
dx
(
u 2 + 2 du
)
(
u 2 + 2 du
)
sin( 3 2x ) dx
1+ x
dx
x
(10) Find
(11) Find
sec (2x + 1) dx
(12) Find
3x 1 dx
(13) Find
x2 1
dx
x2 + 1
2
1
-1-
Math 11B
1
2
(14) Find x + x 3 dx
0
1
4
(15) Find
sin ( 2x ) dx
8
6
(16) Find
tan ( 2x ) dx
8
2
(17) Find
1
2t + 1 dt
1
1
(18) Find e
x
dx
1
(19) Find
xe
(20) Find
4x
(21) Find
(x
(22) Find
tan ( x ) dx
(23) Find
sec ( x ) tan ( x ) dx
(24) Find
x
(25) Find
log( x )
(26) Find
sin( x ) cos ( x ) dx
x2
2
2
dx
sin( x + 2)dx
3
2x ) x 3x + 3 dx
3
2
2x + 1 dx
3 + log( x )
1
dx
x
2
2
(27) Find
x
2
x + 2 dx
3
1
-2-
Math 11B
log( 3)
(e
(28) Find
2e
+ 1)
x
0
2
(29) Find
x
x
4x + 9
2
1
3
e
(30) Find
x cos(2x ) dx
(31) Find
x
(32) Find
x
(33) Find
x log ( x ) dx
(34) Find
log ( x ) dx
(35) Find
log (
(36) Find
(37) Find
x
2
sin ( 2x ) dx
a
log( x ) dx
( 1)
)
x dx
1
log
x
2
( x ) dx
1
tan ( x ) dx
x3
dx
x2 + 1
(39) Find
x e
2 3x
(40) Find e
(43) Find
dx
2
(42) Find
4 x 2 +9
1
(38) Find
(41) Find
dx
x
2x +1
3
dx
dx
cos( x 2 ) dx
x3
dx
x+3
x 3 + x2
dx
x 2 3x + 2
-3-
Math 11B
1
dx
4x + 3
(44) Find
x
(45) Find
(x + 2)
2
x 1
2
dx
x3
dx
(46) Find 2
x + 2x + 1
(47) Find
x
2
x+4
dx
+ 2x + 5
x4 x 3 + 5
dx
x2 + 5
(48) Find
(49) Find
x
x3
dx
2
+ 4x
x 2 + 5x + 2
dx
x 2 + 4x + 3
(50) Find
(51) Find
x
2
(52) Find
x
2
x3
dx
+ 8x + 16
x3
dx
+ 4x + 7
2
(53) Find
x log ( x ) dx
1
(54) Find
log ( x ) dx
(55) Find
tan ( x ) dx
(56) Find
sin x dx
(57) Find
cos x dx
(58) Find
sin ( x ) cos ( 2x ) dx
1
2
2
-4-
Math 11B
(59) Find
sin ( x ) sin ( 2x ) dx
(60) Find the area bounded by y = cos( x ) , y = 0 from x = 0 to x =
.
2
(61) Find the area bounded by y = sin( x ) , y = cos( x ) from x = 0 to x =
.
2
(62) Find the area bounded by y = sin( x ) , y = cos( x ) , y = 0 from x = 0 to x =
.
2
(63) Find the area bounded by y = x 4 , y = 2 x 2 .
(64) Find the area bounded by y = log( x ) , y = log( 4 x ) , y = 0 from x = 1 to x = 3.
(65) Find the average value of sin(x) on 0,
2 (66) Consider the area bounded by y = cos( x ) , y = 0 from x = 0 to x =
.
2
Find the volume of the solid obtained by rotating this area about the x-axis.
(67) Consider the area bounded by y = x 2 , y = x .
Find the volume of the solid obtained by rotating this area about the x-axis.
(68) Consider the area bounded by y = x 2 , y = x .
Find the volume of the solid obtained by rotating this area about the y-axis.
Use the definition to determine whether or not each improper integral below is
convergent and find its value if it is convergent
(69)
1
(70)
1
(71)
1
x
1
x
2
3
1
x
1
3
2
p
dx
dx
is convergent
dx = is divergent
p >1
p 1
-5-
Math 11B
(72)
xe
2x
dx
0
(73)
xe
2
x
3
dx
1
(74)
0
1
x
2
(75)
0
2
3
dx
1
dx
2 x
1
(76)
log ( x ) dx
2
1
(77)
1
dx
x +1
(78)
e
x
dx
0
Use the comparison rule to determine whether or not each improper integral below is
convergent (do not try to find its value)
(79)
e x
2
dx
0
(80)
x
0
(81)
0
4
1
dx
+1
1
1+ x 2 + e x
dx
1 + cos ( x )
dx
x2 + 1
0
(82)
Calculate integrals below
-6-
Math 11B
(83) Find
x ( 3x + 2 )
1
3
(84) Find
x cos(2x
+ 5)dx
(85) Find
cos ( x ) dx
(86) Find
x sin(1 2x ) dx
2
dx
sin( x )
3
(
)
(87) Find cos 2x + 1 dx
(
(88) Find log
(89) Find
x
(90) Find
(x
)
x 2 dx
sin( x + 2)dx
3
2
2
1
+ 2x ) tan ( x )dx
x2 + x
dx
x+2
(92) Find
x 3 + 4x 2
dx
x 2 + 3x 4
(93) Find
x
(91) Find
(94) Find
2
x +1
dx
+ 6x + 9
x 2 + 7x + 10
dx
x 2 + 6x + 12
-7-