Definite Integration with u-Substitution - Homework

Definite Integration with u-Substitution - Homework
S)*>!(42!;7&'2+!%9!(42!9%&&%3)*6!>29)*)(2!)*(2657&+0!k25)9Q!'+)*6!Q%'5!=7&='&7(%50!$%?2!3)&&!'+2!0,+':+()('()%*X
%(425+!3)&&!*%(0
! "x
-0!
%3
4
2
# 1# dx
3
/0!
! x"
#2
#
x # 1 dx
0
%
4
'x4
*
) # x,
!!! ( 4
+#2
1
!!!!!!!
2
24
64
#8#0 !
5
5
4 # 2 # " 4 ( 2# ! #4
% 12
1
! "1 # cos 2 x # dx
.0!
! 2 x" x
2
3
! 2 sin 2 x dx
u ! 2 x, du ! 2 dx
0
2% 3
! sin u du
1
#
2
% 12
x0
2% 3
1
1 1 3
"# cos u# 0 ! ( !
2
4 2 4
3
2
( 1# dx
E0!
% 1
#
12 2
2
| u ! 2 x, du ! 2 x dx
1
! 2 x" x
0
%6
! cos u du
2
2
2
( 1# dx !
0
| x ! 0, u ! 0 | x ! % 12 , u ! % 6 !!
12
#1 1
#2 x "9 # x 2 # dx
!
2 0
! u du
2
0
!! x ! 0, u ! 9 | x ! 3, u ! 0
x ! 0, u ! 1 | x ! 1, u ! 2
0
9
2
- u3 0
8 1 7
/ 2 ! # !
. 3 11 3 1 3
5
#1 0 1 2
1 - 2u3 2 0
! u du ! 2 /. 3 21 ! 9
2 9
0
3
4
!
x # 4 dx
G0!
!
0
x # x dx
I0!
2
3
!
5
! "4 # x # dx ( ! " x # 4# dx
!"
0
4
0
5
4
'
*
x2* 'x2
)4 x # , ( ) # 4 x,
2 +0 ( 2
(
+4
25
17
16 # 8 (
# 20 # "8 # 16# !
2
2
4
!
0
! 2"2t ( 1#
9
1
! " x # x # dx
4
1
#1 2
#1 2
1
!!!!!!!!! 2
2
# 3#
1
2
t sin t dt
u ! cos t, du ! # sin t dt
0
9
du ! " u1 2 # ! 3 # 1 # 2
1
dx !
dx
5
12
3
! 2 x" x
2
#2
# 3# dx
2
6
6
#1 1 5
1 - #10
! u du ! 2 /. u 21 ! 12 ( 2 ! 12
1
1
#2
%2
3
# ! # cos3 t sin t dt
dt
2
2
x ! 2, u ! 1 | x ! 3, u ! 6
2 1
16 - 1 2 0
# ( 8 # #/ # 2 ! 3
3 2
3 . 2 31
! cos
" x 2 # 3#
u ! x 2 # 3, du ! 2 x dx
1
'2 x 3 2 x 2 * ' x 2 2 x 3 2 *
# , () #
!!!!!!!!!! )
,
2 +0 ( 2
3 +1
( 3
"x
-/0!!
0
0
!u
#
%2
4
2
4
x # x dx (
--0!
t ! 0, u ! 1 | t ! 4, u ! 9
1
2
1
x
x
%2
dt
2t ( 1
u ! 2 t ( 1, du ! 2 dt
1
2
!
0
4
9 # x 2 dx
u ! 9 # x 2 , du ! #2 x dx
u ! x ( 1, du ! 2 x dx
%6
% 1
% 1
# "sin u# 0 ! #
12 2
12 4
-D0!
!x
0
%
12
! 2 cos 2 x dx
x ! % 3, u ! 2 % 3
x ! 0, u ! 0
0
0
0
F0!
1
2
'2 x 5 2 x 2 *
# ,
)
2 +0
( 5
2
B0!
! sin"2 x # dx
C0!
0
! t sin"% # t # dt
2
0
2
u ! % # t , du ! #2 t dt
1
#
2
%2
! 2t sin"% # t # dt
2
0
t ! 0, u ! 1 | t ! % 2 , u ! 0 !!!!! t ! 0, u ! % | t ! % 2 2 , u ! % 2
1
0
%2
- u4 0
1
3
%2
1
1
1
1
# ! u du ! / 2 !
#
!
sin
cos
u
du
u
! "0 ( 1# !
"
#
!
%
4
4
.
1
1
2
2 %
2
2
0
!!!"#$%&'(#$&)#'*&+(",+- !! ! ! "#! $%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-.G!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!. /&0. /,)!$(&1
% 3
%4
-C0!
!
2
tan x sec x dx !!!!!!!!!!!!!!!!!!!!
!-B0!
1
! cos x 1 # cos2 x dx !!!!!!!!!!
-.0!
0
0
!x
ax 2 ( b dx
0
% 3
2
u ! ax ( b, du ! 2 ax dx
! cos x sin x dx
1
2a
0
1
! 2ax ax 2 ( b dx
u ! tan x, du ! sec x dx
u ! sin x, du ! cos x dx
0
x ! 0, u ! 0 | x ! % 4 , u ! 1 !!!!!!
!!! x ! 0, u ! b | x ! 1, u ! a ( b
x ! 0, u ! 0 | x ! % 3, u ! 3 2
1
1
3
2
32
- 2u 0
a (b
3 2
2
3 2
ax 2 ( b# # b 3 2
- u2 0
- 1 0- 2 u 3 2 0
"
1 a (b 1 2
3
! u1 2du ! /. 3 21 ! 3
! u du ! /. 2a 21/. 3 21 !
! u du ! /. 2 21 ! 8
0
0
2
3a
a
b
0
b
0
2
%2
4
sin x
dx
-E0! !
x
%2 4
u ! x , du !
-F0!
! 9# x
4
2
dx
-G0!
2 x
3
sin x
dx
2 !
x
2
2
% 4
4
! "9 # x 2 # dx (
! "x
0
3
2
# 9# dx
3 3
4
!!!!!!!!!!!!! '
*
x * 'x3
)9 x # , ( ) # 9 x,
x ! % 2 4 , u ! % 2 | x ! % 2, u ! %
3 +0 ( 3
(
+3
%
%
2 ! sin u du ! "#2 cos u# % 2 ! 2
27 # 9 (
%2
2
!
f " x # dx !
0
-I0!!
!
f " x # dx !
#2
2
M9!
!
f " x # dx !
0
/B0!
!
#2
0
f " x # dx !
!
/D0!!
f " x # dx !
#2
11
and
3
0
22
3
0
2
! 3 f " x # dx ! 11 !!/C0!
! f "3x # dx ! 5
#2
0
0
2
!
2
f " x # dx ! 0
#2
6
f " u# du !
/E0! ! # f " x # dx !
0
1
3
#11
/F0!
3
0
! 3 f "3x # dx
0
1
"15# ! 5
3
2
! 3 f " x # dx ! #11 !/G0!
! f "3x # dx ! 0
#2
#2
2
/C0! x ! 0, u ! 0 | x ! 2, u ! 6
0
0
#11
//0!
3
! f " x # dx ! 15, f " x # !)+!7*!%>>!9'*=()%*!<+Q??2(5)=!(%!(42!%5)6)*@X!9)*>!(42!9%&&%3)*6N
u ! 3 x du ! 3dx
!
2
! # f " x # dx !
/-0!
6
#11
!!/.0!!
3
1
3
64
64
# 36 # "9 # 27# !
3
3
! f " x # dx ! 15, f " x # !)+!7*!2;2*!9'*=()%*!<+Q??2(5)=!(%!(42!6!,!7R)+@X!9)*>!(42!9%&&%3)*6N
2
11
3
1
is not continuous at x ! 0,
x2
!!!!! the definite integral which represents
area under the curve makes no sense.
Since
6
11
and
3
0
dx !<#2!=7529'&l@
9 # x 2 ! 0 4 x ! )3
dx
%2
M9!
2
#4
0
1
1
!x
u ! 3 x du ! 3dx
1 2
! 3 f "3x # dx
3 #2
/G0! x ! #2, u ! #6 | x ! 2, u ! 6
1 6
1
f " u# du ! "0# ! 0
!
3 #6
3
!!!"#$%&'(#$&)#'*&+(",+- !! ! ! "#! $%&'()%*+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!-.I!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!. /&0. /,)!$(&1