!

!"#"$%&"!'"()'"(%*)!%%!

"#$%&'()*!*!+(,(-$.('!!

/'0%#1!234!23(5(#*!!

!!

!

!

!

!

!

6789":7/;!<=!6>9?"<"@!=!/7AB;"A!@>C!

!

!

!

2

3

3( 2) .x x dx ( 1) .x x dx 34 .x x dx

.dxx x

4

5

21 .x

x ee dxx

6

7

2

23 1 .

1x dxx

8

9

2sin 3cos .2x x x dx

2

3

4

,

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

!"#!$%&'()*+*!*!,-%./*0%'1'!)23*0%24'!!! !!!"#$%&'()'&$#*$+&,-./&+0#

5#! 1 1 2!"

" #! ! ! ! ! ! ! 6#! 1 1 2

!"# "

!

!"#$%&#'()$

5## 1 13 3

4 54 5% &

" # " # " # " #" ##

# 3 4 5 4 5% " # & " # #

# 3 4 5 4 5% & " % & # #

6&--&+#(*(+&7*0#

#8

4 5 3% &% & #

2#

9.(#:,%*;'%&(#7&--&+# 3 3< 21 1

% &# #

#

9-.$;.%=#

# 1 13 3 4/$> > /$> >51 1

!" !" !" " # " ## " # " # #" #

#

# 3 /$ 21

" # '# " #

#

6##?,%.('"7=#:,%*;'&+#;&+@-.#12#A&:B/+.+*#"('+"# 3 /$ 21

# " '# # "

#

*+,-.')/0.)12&,.$3+))!"#$%&'()'&$#*$+&,-./&+0#

5#! 1 23!""

!! ! 6#! 11

2!"" (#

! ! 7#! 1 2C 1!""

! ! "#! 1 2D 3E!"

"!

!

!"#$%&'(!)!)*'+',#-'&)2

7#!!"#('F@-+)'&+#;'&#+"#$%&'()'&+#*$+&,-./*0#

# 1 = 82!" ##" (" )

#

!"#$%&#'($

!-*$)7*# 1#" (" ) #('$;"--)'&+#$"#G)-7"$#H.$)$*H&#

#1 11

11

I =1 1I( #) ( (#" (" ) # " # " *# ##

#

HB#

#1

1I 2I#) (*#

#

6&--&+#:"J&$;"(*7*# = 21(" + !" !+#

#K&7*0#

# 1 13 2!" !+,##" (" ) + *

#

L.-"(*('+=#$,.#J/&-.#F"#7&--#*#;.//)%7"#-.(+&+0#

35# 8<* #1

1IM272+'2 8 =I#) (#

#.+"'&-"#

# 11

3 3 11

!+, + !+ '# # #" (+

#

15# 8<* #1

1IM272+'2 8 =I#) (#

#.+"'&-"#

1 1 1 1

3 3 3 1 1.-N+.$ .-N+.$I I

!+ + #" (, '# # * *+ * #) ( #) (

#

O5# 8<* #1

1IM272+'2 8 =I#) (#

#.+"'&-"#7&--&+#

1 1 1

3 3 3 3 /$1

!+ !+ + *, '# # # *+ * + *+ *

#

'"'(!.')/'#$/'#!,$)!!)

3

###1

1 1

3 1 I/$ 2I 1 I

#" ( ( #) '( #) #" ( ( #)

#

!"#$%&'()'&$#*$+&,-./&+0#

"#! 1 2E 3P!"

" "!! ! ! ! 8#! 1 2

D E!"

" "# # # # 9#! 1 2

I I!"

" "#

:#! 1 2I

!"" "

# # # # # # ;#! 1 23

!"" "

# # # # <#! 1 2O 1 1

!"" "

#

!"#$%&#'()

"#! 1 11 1 3 E.-N+.$

E 3P 4 O5 C I 3 3P OE I 3 3P OE!" !" "

" " "!

! 1 1 E 3 O.-N+.$ .-N+.$ 2E O OOE" " ' !

4&05$3##Q$+&,-./*$#&#;'"$"#$"#;&+@-"$#&#GB$;*+#;)#+"#7B$;#+.#:,%*;'*7#&;'&#7&#.$"#+"#7&+);"(#("#:"J&$;"(*7*+0#

# 1 1 1 1

O O O O 3 .-N+.$OO4 O5 O C C C4 35

" -!" !- !- -!" !-" - -

#

# # # # # ## 3 O.-N+.$ 2O O

" ' #

8#!?"#H"+"#-.(+0# 3= D= E2# ( ) #

K&7*#1

1I I 3 E 1D 3 2

I II#) (*#

#

9-.=#H&7*#-.(+*$#&#+-&+"0#9-.$;.%#

#1

1 1 1

1 3 D 4 D5 I 3 E3 /$D E 4 D5 I 3 E 1 3 D 4 D5 I 3 E

"!"," " "

#

# ### 1 E O/$ /$ 21 I 1" " '" "

#

4&05$3##M&+@-"$#7B$;#+.#:,%*;'*7#&;'&#(*#J*%)$0#

!"#$%&'(!)!)*'+',#-'&)4

14 15 4 O5

4 154 O5 4 154 O5D E!" !" " ", !"

" " " "" "#

### O/$> O> /$> 1> /$ 2O 1 1

!" !" "" " ' '" " "

#

9##R&'+"#J"-&%7"#(&# 8 0* #

M272+'2#

# 1 11 3 2

1 3 I 1I I 4 15!" !" ' '

" "" " "#

:##K&7*# 3= I= 82# ( ) #S+"'&-"# I 82* #

9-.$;.%#

# 13 1 I I 3/$ /$ 2I 1 I I I II

!" " " '" "" "

#

;#! 1 11

3 O1 1

3 O3 O11 1

" -!" !"" "

!" !-"

#

# 1 1 11

3O O 3 1 O 1.-N+.$O1 1 O3 OO O

I 11 1

"!- !--

-

#

# 1 O 1 3.-N+.$ 2O O

" ' #

<#! 111

33 3 1 31 1 OO 1 1 1O O C O CO O

!" !" !""" " " ""

#

1 111

3 D D3 3O O OO OD3 D D 4 35OO O O

" - !-!"

!" !-" -

#

'"'(!.')/'#$/'#!,$)!!)

5

# 3 O O 3 3 O 3.-N+.$ .-N+.$ 2O D D D D

" " ' #

)*+,-.')/0.)12&,.$3+))!"#('F@-+)'&+#;'&#+"#$%&'()'&+#*$+&,-./*#*#G)-7"(0##

8#! 1 2!"" (" )

! ! ! ! ! ! ! ! 9#! 1 2!"#" " )

!

!"#$%&'()'&$#*$+&,-./&0#

:#! 1 2I I!"

" "## # # ;#! 1 2

D 3!"

" "## # # <#! 1 2

O I!"

" "#

5=#! 1 23

!"" "

# # # # 55#! 1 23

!"" "

# # # # 56#! 1 2O D

!"" "

!

?"# -.(+*$# &# T"-,%*+'('"7=# *$+&,-./*# *# GB$H(*)$&J&# -.N*)$./&#('$;"--)'&+#$"#*$+&,-*7*$#&#+'@&("(#("#+-.%+"(#

# # # # # # # # # 4 54 5. "/ "

## # # # # # # # # # # # 435#

HB## ;&, 4 5 ;&, 4 52. " / " #

?"(&#T)/*$)7*# 4 5/ " #7B$;#+"#('T-&'&+#(*0#14 5 4 5 4 50 1/ " " # " ." / #

HB# ## "('+"# --"$%"# -&./&# &#0# U# G*('+# &# T)/*$)7*+=# HB-(&# 2 # %.$"#--"$%"# +"# ('B7"G*('+.# H)7T/&H(&# .+"'&-"# ('T-&'%.# 435# T.-.F*+&+# $"#+-.%+"$0#

3 1 3 31 1 1

4 5 222 222 24 5 4 5 4 5 4 5

0 1 10 1

% & ' "% % & ' ". "/ " " # " # " # " ." / " ." /

#

!"# T.$%)'B-.+# 3 3 3=222= = =222= = =222=0 1 1% % & & ' ' # N.H+)'&$# 7&# 7&+);"$# &#H)&G*N*&$+&J&#+"#T.N.H+B.-2#!"#$%&'()'&$#*$+&,-./&+0#

5=#! 24 154 O5

" !"" "

!! ! ! ! ! 55#!1

11 2

4 35 4 15" !"

" "#

!"#$%&'(!)!)*'+',#-'&)6

56#!1

I1 2

3" !"

"# # # # # # # # 57#! O 2

3!""

#

!"#$%&#'()

5=#!VB$H(*)$*$!4 154 O5

"" "

!&#('"$)%7"#$"#+-.%+"$#

# 24 154 O5 1 O

" % &" " " "

## # # # # # # # 435##

MBH&#('B7":B.-#+"#;@#.$"+#&#435#7&# ( 2)( 3)x x #7&--&+0#

( 3) ( 2)x A x B x #

( ) (2 3 ).x A B x B A #

9-.=#7&--&+#(*(+&7*0#

#1

2 3 0A BB A

#

7&#:,%*;'%&$#&#+"#N*/*+#H&7*0# 2 3, .5 5

A B #

9-.##

# 2 3 2 3ln | 2 | ln | 3 |( 2)( 3) 5 2 5 3 5 5

x dx dxdx x x

x x x x#

# # # # # # ### 2 3 2 31 1 1ln( 2) ln | 3 | ln( 2) | 3 | .5 5 5

x x x x C #

55##MBH&#J&T-B.-#$,%.('"7#7&--&+0#1

1 11

3 14 35 4 15 4 35" % & '

" "" " "#

2 22 ( 1)( 2) ( 2) ( 1) .x A x x B x C x #

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

13 2 02 2 2

A CA B CA B C

#

6&#:,%*;'%&$#&#+"#N*/*+#7&--&+0# 5, 3, 6.A B C #

'"'(!.')/'#$/'#!,$)!!)

7

9-.=#1

1 11 D O E

3 14 35 4 15 4 35" !" !" !"!"

" "" " "#

# # # # # # ## 35ln | 1| 6ln | 2 |1

x xx

#

# # # # # # ## 6 5 3ln( 2) ln | 1|1

x xx

6

5

( 2) 3ln .| 1| 1x

Cx x

56#!1 1 1

I 1 1 11 1 13 4 354 35 4 354 354 35

" " "" " " " " "

#

9-.#

#1

1 11

3 34 354 354 35 3" % & '" 3

" "" " " "#

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

0200

A B CA B DA B CA B D

6&#:,%*;'%&$#&#+"#N*/*+#H&7*0# 1 1, , 0, 1.2 2

A B C D #

9-.#

#2

4 2

2 1 11 2 1 2 1 1

x dx dx dxdx

x x x x#

# # # # ## 1 1ln | 1| ln | 1| arctan2 2

x x x #

# # # # ## 1 1ln arctan2 1

xx C

x2#

!"#$%&'(!)!)*'+',#-'&)8

57#! 3 2 2

1 111 ( 1)( 1) 1A Bx Cxx x x x x x

!

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

00

1

A BA B CA C

!

6&#:,%*;'%&$#&#+"#N*/*+#7&--&+0# 1 1 2, , .3 3 3

A B C #

M272+'2#

13 2

1 21 13 3 ln | 1|3 1 31 1

xdx dxdx x I

xx x x#

HB#

# 1 2 2 2

1 21 2 1 1 2( 2)3 33 3 21 1 1

x x xI dx dx dx

x x x x x x#

# ##### 2 2 2

1 2 1 3 1 2 1 16 6 21 1 1

x x dxdx dx

x x x x x x#

# # # 2 31 1 .6 2I I #

# 22 2

2 1 ln( 1).1

xI dx x x

x x#

# 3 2 2

11 2 2 12arctan arctan .

1 3 3 3 31 32 22 2

xdx dx x

Ix x

x

#

9-.#

# 21

1 1 2 2 1ln( 1) arctan .6 2 3 3

xI x x #

M272+'2##

'"'(!.')/'#$/'#!,$)!!)

9

23

1 1 1 2 1ln | 1| ln( 1) arctan3 61 3 3

dx xx x x

x#

# # ### 2 21 1 1 2 1ln( 1) ln( 1) arctan6 6 3 3

xx x x #

# # ###2

2

1 ( 1) 1 2 1ln arctan .6 1 3 3

x xC

x x#

!"#$%&'()'&$#*$+&,-./&+0#

5"#! 4

1 .1dx

x!! ! ! ! ! ! ! 58#!

O 1

OD O 1 2

D" " " !"

" "#

59#!5 3

3

3 .x x xdx

x x# # # # # # 5:## 3 2

2 .1

xdx

x x x#

5;## 5 4 3 2 .1

dxx x x x x

# # 5<#! 3 .3 2dx

x x#

6=## 2 2 .(1 )(1 )(1 )

dxx x x x

#

!"#$%&#'()5"##6%&07$32#W'T-&'%.#&#;'"$"#+-.$(G)-7)'&+#(*#J*%)$0#

4 4 2 2 2 2 2 2 2

1 1 1 11 2 1 2 ( 1) ( 2 ) ( 1 2 )( 1 2 )x x x x x x x x x x

#

# #2 2

1 .( 2 1)( 2 1)x x x x

#

M272+'2##

4 2 2

1 .1 2 1 2 1

Ax B Cx Dx x x x x

#

58##W"#T.-*#H-@&%7"#T%&("+*7*$2#6&--&+##O 1 1

O 1D O 1 D 1 13

D 4 D5" " " " "

" " " "2#

!"#$%&'(!)!)*'+',#-'&)10

MBH&#J&T-B.-#(*#$"#;&+@-.+#T.-.T-.H&#7&--&+0#1

1 1D 1 1 24 D5 D" " % &" '

"" " "#

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

#5

25 2

A BCA

#

T-&%#$,.# 2 23, , 2.5 5

A B C #

M272+'2##

O 1

O 1

1 1O 1D O 1 D D3D D

"" " " !" !""" " "

#

11 1O 1/$> > .-N+.$D D D D D

" "" " !""

!

11 1O 1/$> > /$4 15 .-N+.$ 2D 38 D D

"" " " ' !

59#!L&T-)%7"!$,%.('"7#(*#$"#;&+@-"$#T.-.T-.H&0#

#D O

1O O

O OO I" " " """ " " "

##

M272+'2#D O

1O

O OO I4 354 35

" " " "!" " !"" " "" "

#

# # # # # # O O1

O 3I O I /$ 21 33

!" "" " " " '""

#

5:#! O 1 1 11 1 1 2

3 4 35 4 35 4 35 4 35" " "!" !" !"

" " " " " " " "#

MBH&#J&T-B.-#(*#$&#;&+@-.+#T.-.T-.H&#7&--&+0#

'"'(!.')/'#$/'#!,$)!!)

11

1 11 2

3 34 35 4 35 4 35" % & '

" "" " "#

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

31 8 =

1 1

% '& '% & '

#

7&#:,%*;'%&$#&#+"#N*/*+#7&--&+# 3 O O= = 2I 1 I

% & ' ##

9-.##

1 11 3 O O

I 3 1 I 34 35 4 35 4 35" !" !" !"!"

" "" " "#

# # # # # # ### 3 O 3 O/$> 3> /$> 3>I 1 3 I

" ""

#

# # # # # # ### O3 O/$> 3>> 3> 2I 14 35

" " '"

# #

5;##6&F"#D I O 1 I 13 4 35 4 35 4 35" " " " " " " " " " #

# # # # I 1 I 1 14 354 35 4 354 1 3 5" " " " " " " #

# # # # 1 1 1 14 3544 35 5 4 354 354 35" " " " " " " " #

7&--&+0#

5 4 3 2 2 2

11 1 1 1

A Bx C Dx Ex x x x x x x x x x

#

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

02 02 2 0

2 01

A B DC D EA D EB D EA C E

#

!"#$%&'(!)!)*'+',#-'&)12

7&# :,%*;'%&$# &# +"# N*/*+# 7&--&+0# # 1 1 1, 0, , ,3 2 3

A B C D #

1 .6

E #

9-.##

5 4 3 2 2 2

1 11 1 3 6

1 3 1 2 1 1

xdx dx dx dxx x x x x x x x x x

#

21 1 2 1 1ln | 1 | arctan ln( 1)3 63 3

xx x x #

2

2

1 ( 1) 1 2 1ln arctan .6 1 3 3

x x Cx x

#

5<##W"#T.-*#G.H+)-*:)%7"#('T-&'%&$# 3 23 2 ( 1) ( 2).x x x x #

M272+'2#

# 3 2

13 2 1 ( 1) 2

A B Cx x x x x

#

9.(#+-.$(G)-7*7&J&#7&--&+#(*(+&7*0#

##02 0

2 2 1

A CA B CA B C

#

7&#:,%*;'%&$#&#+"#N*/*+#7&--&+0# 1 3 1, , .5 5 5

A B C #

M272+'2#

# 3 2

1 3 13 2 5 1 5 ( 1) 5 2dx dx dx dx

x x x x x#

# 1 3 1 1 1 2 3ln | 1 | ln | 2 | ln .5 5 1 5 5 1 5( 1)

xx x Cx x x

#

!!

'"'(!.')/'#$/'#!,$)!!)

13

6=## 6%&07$3()2 3 2 2(1 )(1 )(1 ) (1 )(1 )(1 )(1 )x x x x x x x x #

# # # # # # ## # 2 2 2(1 ) (1 )(1 ).x x x x #

M272+'2#H&7*0#

# 2 2 2 2 2

1 .(1 )(1 )(1 ) 1 (1 ) 1 1

A B Cx D Ex Fx x x x x x x x x

#

)*+,-.')/0.)12&,.$3+))!"#$%&'()'&$#*$+&,-./&0#

57#! I 3E!"

"! ! ! ! ! ! ! 5"#!

O 1

OO 1 3 2" " " !"" "

!

58##5 3

3

2 2 .2

x x xdx

x x## # # # 59#!

5

4

2 .xdx

x x## #

5:#! 3 2

2 .3 3

xdx

x x x## # # # 5;## 5 4 3 2 .

1dx

x x x x x# #

5<## 3 2 .1

dxx x x

# # # # # 6=#! 2 2 2 .(1 ) (1 )(1 )

dxx x x x

!

>'&0?2!'!@/&)0()2?/.*&!R&# +"# %&$"# ( )P x # ;'&# ( )Q x # T)/*$)7&#7&# H)&G*N*&$+# -&./"2# R&# +"# %&+"#('H.//.#&#T)/*$)7*+ ( )P x #7"#&#J),"/#(&#('H.//.#&#T)/*$)7*+# ( ),Q x #T-.#

+'@&(.# ( )( )P xQ x

##"('+"#&#--&,B//+2#

435#R&#+"#%&+"#

# 1 12 21 1 1( ) ( ) ... ( ) ( ) ... ( ) sr

r s sQ x x a x a x p x q x p x q #

HB# , ,i i ja p q #%.$"#$B7-.#-&./#+"#+*//"#F"#2

0,4j

j

pq #HB-(&# i #;'&# j #

%.$"#$B7-.#$.+@-)-=# 1, 2,..., ; 1, 2,..., .i r j s #

R&#+"#%&$"#

!"#$%&'(!)!)*'+',#-'&)14

1 1 11 1 12 21 1 1 1( ) ( ) ... ( ) ( ) ... ( ) sr

r s sQ x x a x a x p x q x p x q2 2

2 1 1 1( ) ( ) ... ( ) ( ) ... ( ).r s sQ x x a x a x p x q x p x q #

S+"'&-"#&H:*(+)%$"#T)/*$)7&+# 1( )P x #;'&# 2 ( )P x #('H.//.#&#+"#N*/"J&#"('+"#7"#&#J),"/#(&#('H.//.#&#T)/*$)7*+# 1( )Q x #;'&# 2 ( ),Q x #HB#T)#&#N&H*7#F"#('H.//.# &# T)/*$)7*+# 1( )Q x # "('+"# 1n # HB-(&# ('H.//.# &# T)/*$)7*+# 2 ( )a x #"('+"# 2 ( 2 ),n r s #+"#+*//"#F"#

1 2

1 2

( ) ( )( ) .( ) ( ) ( )

P x P xP x dx dxQ x Q x Q x

#

K)&G*N*&$+"+# &# T)/*$)7&J&# 1( )P x # ;'&# 1( )Q x # $%&'()'&$# 7&# .$"# +"#7&+);"(#("#H)&G*N*&$+&J&#+"#T.N.H+B.-2#

ABC%*+*!5##9)/*$)7&+# 1( )Q x #;'&# 2 ( )Q x #7B$;#+"#N.H+)'&$#$"#7"$@-"#+"#;-&%+T"-;-&%+"#&;'&#T.#&#;*+B-#:X"-+'*7*$#435#&#T)/*$)7*+# ( )Q x #$"#G.H+)-"#+"#T.:X"-+'@&('"72#

9)/*$)7*# 1( )Q x # "('+"# T%&("+B&(*# 7"# *# 7.;'"# *# T"-X.('H"+# *#T)/*$)7&J&# ( )Q x # ;'&# ;&-*J.+*+# +"# +*%# ' ( ),Q x # ;'&# 7B$;# +"# N.H+)'&+#

T"-7&(#./,)-*+7*+#+"#YBH/*;*+=#,%&-"(.# 21

( )( ) .( )

Q xQ xQ x

#

ABC%*+*! 6#!K%)#7&+);"# ("# T.-*# B# TBX/*HB.-# $,.#62L2#Z(+-),-.;(H*#7"#3PID#;'&#T"-#'*-"#+"#+*%#7X.$"#H"+"#&7"-2##MBH&#:X.+B.-#7&+);"$#&#Z(+-),-.;(H*+#+"#$%&'()'&$#*$+&,-./&+0#

68## 2 3( 1) ( 1)xdx

x x# # # # # 69#! 3 2 .

( 1)dxx

#

!"#$%&#'()68##?"#-.(+*$#+)$"0#! ! ( )P x x !

! ! 2 3( ) ( 1) ( 1) .Q x x x #

S+"'&-"# 2 1 3 1 21( ) ( 1) ( 1) ( 1)( 1)Q x x x x x #

'"'(!.')/'#$/'#!,$)!!)

15

# 1deg ( ) 3.Q x #

# 2 2( ) ( 1)( 1) deg ( ) 2.Q x x x Q x #

M272+'2# T)/*$)7&+# 1( )P x # ;'&# 2 ( )P x # .('+B# F"# 1deg ( ) 2;P x #

2deg ( ) 1.P x #

9-.# 21 2 1 1( ) ; ( ) .P x Ax Bx C P x D x F #

S+"'&-"0#2

1 12 3 2( 1) ( 1) ( 1)( 1) ( 1)( 1)

D x Fxdx Ax Bx Cdx

x x x x x x#

?"(&#JT&-)%7"#(*#$"#-.(+&+#T.-.T-.H&#7&--&+0#

1 1

( 1)( 1) 1 1D x E dx dxdx D Ex x x x

#

9-.=#H&7*0#2

2 3 2 .( 1) ( 1) ( 1)( 1) ( 1) 1

xdx Ax Bx C dx dxD E

x x x x x x#

M*G&-&$N)%7"#+"#;@#.$"+#&#('T-&'%&(#("#GB$;*+=#7&--&+0#2 2

2 3 2 3

( 1)(2 ) (3 1)( )( 1) ( 1) ( 1) ( 1)

x x Ax B x Ax Bx Cx x x x

#

.1 1

D Ex x

#

MBH&#J&T-B.-#(*#$"#-.(+&+#&#7"(*T"-7&#7&--&+#(*(+&7*#*#X.-.:*7&J&0#

#

02 0

2 2 02 _ 3 2 1

0

D EA DA B EA B C D

C B D E

#

!"#$%&'(!)!)*'+',#-'&)16

9.(# :,%*;'%&(# ("# (*(+&7*+# 7&--&+0# 1 1 1; ; ;8 4 16

A B C D #

1 .6

E #

M272+'2#2

2 3 2

2 1 1ln ; 1.( 1) ( 1) 8( 1)( 1) 16 1

xdx x x x C xx x x x x

#

69## ( ) 1P x #

# # 3 2 2 2 2 2 2( ) ( 1) [( 1)( 1)] ( 1) ( 1) .Q x x x x x x x x #

S+"'&-"#

# 2 1 2 2 1 21( ) ( 1) ( 1) ( 1)( 1).Q x x x x x x x #

# 1deg ( ) 3.Q x #

# 22 2( ) ( 1)( 1). deg ( ) 3.Q x x x x Q x #

9-.=#&H:*(+)%$"#T)/*$)7&+# 1( )P x #;'&# 2 ( )P x #.('+B#F"0#

# 1 2deg ( ) 2; deg ( ) 2.P x P x #

9-.# 2 21 2 1 1 1( ) ; ( ) .P x Ax Bx C P x D x E x F #

K&7*0#

# #22

1 1 13 2 2 2( 1) ( 1)( 1) ( 1)( 1)

d x E x Fdx Ax Bx Cx x x x x x x

#

# #2

3 2 3 2 .( 1) 1 1 1dx Ax Bx C dx Ex F

D dxx x x x x

#

M*G&-&$N)%7"#+"#;@#.$"+#;'&#:,%&;'*7#(*(+&7*$0#6&--&+0#

# # 1 2 2 40; ; ; ; .3 9 9 9

A C B D E F #

M272+'2#

'"'(!.')/'#$/'#!,$)!!)

17

# # 3 2 3 2

2 2 2ln | 1 |( 1) 3( 1) 9 9 1dx x xx dx

x x x x#

# #2

3 2

1 ln( 1) 2 2 1arctan , ( 1).3( 1) 9 1 3 3 3x x x

C xx x x

#

#*+,-.')/0.)12&,.$3+))MBH&#:X.+B.-#7&+);"$#&#Z(+-),-.;(H*+#+"#$%&'()'&$#*$+&,-./&0#

65#! 2 3 .( 1)

dxx x

! ! ! ! ! ! ! ! 66#! 3 3 .( 1)

dxx x

!!

67#!2

2 2 .( 2 2)

x dxx x

# # # # # # # 6"#! 4 2 .( 1)dx

x# # #

68#! 4 3 .( 1)dxx

# # # # # # # # # 69#!2

4 2 2

1 .( 1)

x dxx x

## #

6:#!5

5 2

4 1 .( 1)

x dxx x

# # # # # # # 6;#! 4 3 2 .2 3 2 1

dxx x x x

!

!!#

!"#!$%&'()*+'&!)',-)'%&'!!!!"#$%&'(&'#)*+,-./0#+&1-+&'0&#2"+#'%&345,5'#&#5'0&$+/.&6&7#

.## .n xx e dx # # # # /## (ln ) , 1.a nx x dx a ## # # 0## sin .n xdx #

1## .sinndxx# # # # "## 2 2 , 1.

( )ndx n

x a#

!"#$%&#'(!

.## 1 1

n

n x n n x x nn

x

x n

I x e dx nx dx du x e e nx dx

v e

#

# # 11.

n x n x n xnx e n x e dx x e nI #

/#1

1

(ln )

(ln )(ln )

n

na n

x u

n xx x dx dx dux

v x

##

1 1 1(ln )(ln )1 1

a a nnx n x xx dx

a a x#

# # # #1 1

11(ln ) (ln ) ln .

1 1 1 1

a an a n n

nx n x nx x x dx x Ia a a a

#

0##

1

1 2

sin

sin sin sin ( 1) sin coscos

n

n n nn

x u

I dx x x dx n x xdx duv x

#

# # 8 9 9:*4 45' ; 8< 45' :*4" "# # " # # $# #

# # 8 9 9:*4 45' ; 8< 45' ;8 45' <" "# # " # $# #

!"#$%&'(!)!)*'+',#-'&)2

# # 8 9:*4 45' ; 8< 45' ; 8< =" ""# # " # $# " % #

>+/##8

9; 8< :*4 45' ; 8<"" " "% " % # # " % #

2+&%#'$/#,&++&07#

# 89

8 8:*4 45' ="" "

"% # # %" "

#

1##

2

2 2 1

2

1sin

1 ( 2) cossin sin sin sin

ctgsin

n

n n n n

ux

dx dx n xI du dx

x x x xdx

v xx

#

# # 2 1

ctg cos( 2) ctg

sin sinn n

x xn x dx

x x#

# # 2 1

coscos cossin ( 2)

sin sin sinn n

xx xx n dx

x x x#

# #2

1

cos cos( 2)

sin sinn n

x xn dx

x x

2

1

cos 1 sin( 2)sin sinn n

x xn dxx x

1 2

cos( 2)

sin sin sinn n n

x dx dxn

x x x

21

cos( 2) ( 2)

sin n nn

xn I n I

x

21

cos( 2) ( 2)

sinn n nn

xI n I n I

x

21

cos 2 , 2.( 1) sin 1n nn

x nI I nn x n

#

'"'(!.')/'#$/'#!,$)!!)

3

"##2 2 2

2 2 2 2 2

1( ) ( )n n n

dx x a xI dx

x a a x a#

# 2 2 2 1 2 2 2

1 1( ) ( )n n

dx xdxxa x a a x a

#

# 12 2 2 2

1 1( )n n

xdxI xa a x a

# # # # # # # # # # ;8<#

?&#0"#'%&34*%,"# 2 2( )nxdxx

x a=#@&,5#A"6&'("45,5'#

#

2 2 2 2 1

1( ) 2( 1)( )n

x udx du

xdxvx a n a x

#

>+/#

#2

2 2 2 2 1 2 2 1

1( ) 2( 1)( ) 2( 1) ( )n n n

x dx x dxx a n a x n a x

#

# 12 2 1

1 .2( 1)( ) 2( 1) nn

x In a x n

#

>+/B#(-1&#A"6&'("4-/+#+&A-.0/05'#&#)-'(50#'"#;8<#,&++&07#

# 1 12 2 2 2 1

1 1 12( 1)( ) 2( 1)n n nn

xI I Ia a n a x n

#

# 12 2 2 2 1

1 112( 1) 2 ( 1)( )n n

xIa n a n a x

#

# 12 2 2 1

1 (2 3) .2 ( 1) ( )n n

xn Ia n a x

#

!!!!

!"#$%&'(!)!)*'+',#-'&)4

)*+,-'./0-.12&+-$3*..!"#'%&34*3&'#5'0&$+/.&07#

.#! 122 2 2 2

( 1)sin sin ( sin cos )ax

ax n nn n

e n nI e xdx x a x n a Ia n a n

!! !

/#!1

42

sin cos 1cos .n

n nx x nI xdx In n

! ! ! !

0#!!&<! 12

1tg tg( 1)

n nn nI xdx x I

a n#

# '<# 12

1ctg ctg , ( 1).( 1)

n nn nI xdx x I n

n#

1#!2 2

2 21

( ) 2 1( ) , .2 1 2 1 2

nn n

x a x naI a x dx I nn n

#

"#! 21

1 sin 2 .cos 1 cos 1n nn n

dx x nI Ix n x n

#

2#! 1 2 2

cos 1sin ( 1) sin ( 1)( 2) sinn n n n

xdx x axIax n a x n n a ax

#

########## 22 , ( 2).1 n

n I nn

3#! 1 2 2

sin 1cos ( 1) cos ( 1)( 2) cosn n n n

xdx x axIax n a x n n a x

#

22 , ( 2).1 n

n I nn

!

4#! 21

1 , ( 1).sin cos ( 1) cosn nn n

xdxI I nax ax a n ax

!

5#! 21

1 , ( 1).sin cos ( 1) sinn nn n

xdxI I nax ax a n ax

#

.6##1

2sin sin

.cos ( 1)

n n

n nax ax

I dx Iax a n

#

..#!1

2cos cos

, ( 1).sin ( 1)

n n

n nax ax

I dx I nax a n

!

'"'(!.')/'#$/'#!,$)!!)

5

./#! 1.n ax

n axn n

x e nI x e dx In a

!

.0#! 11

1, 1.

1

ax ax

n nn n

e eI dx aI n

x n x!

.1#!1

2 2

coscos ( cos sin )ax n

ax nn

e xI e xdx a x n xa n

#

###### ## 22 2

( 1), 1.n

n nI n

a n

."#! 1ln ln , 1.n nn nI xdx x x nI n !

.2#! 11

1 , 1.(ln ) ( 1) ln 1n nn n

dx xI I nx n x n

!

.3#! 11

ln ln, 1.

( 1) 1

n n

n nm m

x x nI dx I m

x m x m!

.4#!1

11

1, 1.

ln ( 1) ln 1

m m

n nn n

x x mI dx I n

x m x n!

.5#! 11 1

1 1 , 1.ln ( 1) ln 1n nm n m n

dx mI I nx x x n x n

!

!!

!"#!$%&'()*+*!*!,-%./*0%'1'!*)23*0%24'!!!!"#$%&'()'&$#*$+&,-./&+0#

5#!35.dx

x x! ! ! ! ! ! ! ! 6##

3

1 .2 2x dx

x x! ! !

7#!2

2 2 .4 4 2

x dxx x x

!! ! ! 8#!3

3

2 , 2.2

x x dx xx x

!

9#!4 23

.( 3) ( 5)

dx

x x!! ! ! ! ! "#! .

1 1dxx x

!

:##2

2.

1x dxx x

# # # # # # # ;#!2 2

.( 1) 1

dx

x x x x!

!"#$%&#'(!

5##1.--*2#3"4&$5"(*2*$# 10 9; 10 .x t dx t dt #

6-.#

#9 9 4

5 6 535

10 10 10(1 ) 1

dx t dt t dt t dtt t t t tx x

!

! 3 2 110 11

t t t dtt

!

!4 3 2

1010 ln | 1 | ku .4 3 2t t t

t C t x !

6#!7"4&$5"()%2"# 6 52 , 6 .x t dx t dt #1&--&+0#

#6 5 6 5

3 2 23

1 ( 1)6 ( 1)6( 1)2 2

x t t t tdx dt dtt t t tx x

#

#6 3 9 8 7 6 5 4( 1)6 61 9 8 7 6 5 4

t t t t t t t tdt C

t#89# 6 2.t x #

!

!"#$%&'(!)!)*'+',#-'&):#

7##;$+&,-./*$#&#5'"$"#&#('"$)%2"#$"#+-.%+"$0#

#2

4 32

22 2 2 22 2( 1)2( 2) 2

x tx t tI dx tdt tdt

t t t tdx tdtx x!

! !!! 3 2

22 2 .1 1 1t A Bt Cdt dt

t t t t!

6.(#+-.$(<)-2*2&4&#2&--&+# 1; 1; 1.A B C #

6-.#

# 2

12 21 1

dt tI dt

t t t#

# 2 1 2 12 ln | 1 | ln( 1) arctan ,3 3

tt t t C #89# 2.t x #

8##7"4&$5"()%2"# 3 22 , 3x t dx t dt #

#3 3 2 6 3

3 33

2 ( 2) 3 23( 2) 22

x x t t t t tdx dt dtt t t tx x

!

!2 4 2 2

32 2

2 3 3 23 3( 1)( 2) 4 2 ( 1)( 2)

t t t t t tt t dt dt

t t t t t t!

='>-&'%&$#$"$#<9$8(*)$*$#*$+&,-./#&#+-.$(<)-2)%2"#(*#4*%)$0#

#2

2 2

3 6 .( 1)( 2) 1 2

t t A Bt Ct t t t t t

#

6.(#3,%*5'%&(#2&--&+0#1 5 1; ; .4 4 2

A B C #

6.(#3,%*5'%&(#2&--&+0#

# 4 2 23 3 3 15 27 2 1ln | 1 | ln( 2) arctan ,4 2 4 8 4 7 7

tI t t t t t C #

89# 3 2.t x #

'"'(!.')/'#$/'#!,$)!!)

#?#

9##4 2 2 23

6 23 34

.( 3) ( 5) ( 5) 5( 3) ( 3)( 3) 3

dx dx dxIx x x xx xx x

!

7"4&$5"()%2"0# 3 22

5 3; .3 ( 3) 2

x dxt t dtx x

#

6-.#

#2

32

3 3 3 5 .2 2 2 3t dt xI t C Ct x

#

"##7"4&$5"()%2"0# 1x x u #

@+"'&-"#

# # 22 1 1x x x x u #

# # 22 2 1 1x x x u #

# # 22 ( 1) 1x x x u #

# # 22 1x u u #

# #22 21 1 .

2 2u ux xu u

#

@+"'&-"#

# #'2 21 12

2 2u udx duu u

#

# #2

3

( 1)( 1)( 1)2

u u udx duu

#

@+"'&-"#

# #

2

23

3

( 1)( 1)( 1)1 ( 1)( 1)2

1 2

u u uu uuI du du

u u#

!"#$%&'(!)!)*'+',#-'&)A#

# #3 2

3 2

1 1 1 1 1 1ln2 2 2 2 4u u u du u u C

u u u#

# # 21 1ln( 1) 1 , 0.2 2 2

xx x x x x C x #

:##;$+&,-./*$#&#5'"$"#&#+-.$(<)-2)%2"#(*#4*%)$0#

2 2

2 2 2

11 2

1 1 1

xx x xdx dx dxx x x x x x

#

#

# 2

2 2 2

1 1 2 1 112 2 21 1 1

dx x dxx x dxx x x x x x

#

#2

2 21 3 1 11 ln 12 4 2 2

x dx x x x x x C #

# 2 2 21 2 3 11 ln 1 14 8 2x

x x x x x x x #

# 2 21 1 2 3ln 1 12 2 4

xx x x x x #

# 21 1ln 1 .8 2

x x x C #

;#!;$+&,-./*$#&#5'"$"#&#>.-.B&(*2#$"#+-.%+"$#

#2 2 2 2

.( 1) 1 1 3 1 5

2 4 2 4

dx dx

x x x xx x

!

7"4&$5"()%2"#

#1 ;2

x t dx dt #

'"'(!.')/'#$/'#!,$)!!)

#C#

#2 2

.3 54 4

dtIt t

#

7"4&$5"()%2"#2

,54

tu

t#.+"'&-"#2&--&+0#

#2

1 1 3 2 2ln32 6 3 2 28

dt uI C

ut#

#2

2

3( 1) (2 1) 21 1 5ln , ku .2 26 3( 1) (2 1) 2

x x xC x

x x x#

)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

5#!!343.

( )dx

x x x# # # # # # 6#!

3 27.dx

x x! ! !

7###4

.1 1dx

x x# # # # # # 8#!

3.

3 3xdx

x x# # # # # #

9##3

1 1 .1 1x dx

x x# # # # # # "##

3

121 .

1 131 1

xx

x xx x

#

:##3 .3

x x dxx x

# # # # # # # ;#!4

4

1 2 .( 1) 2x x

dxx x

!

<##1 1

.( ) ( )n nn

dx

x a x b## # # # 5=#!

2

2.

(3 )n

x dx

x x#

55## .1 2 3

dxx x

## # # # 56#!2 2 2 .x x dxx

!

!"#$%&'(!)!)*'+',#-'&)D#

57#!2

2

2 .1

xdx

x# # # # # # # # 58#!

2 2.

( 1) ( 1)dx

x x x x!

59#!2 2

.( 1)( 1)

dx

x x x x! ! ! 5"#!

2 2.

( 1)( 1)dx

x x x x! ! !

>?1'%@?/*+'&!'!A-4')*&#

;$+&,-./*#*#<)-2"(0#

# 2( )I R x ax bx c dx # # # # # # # # # # # EFG#

89#"#"('+"#<9$8(*)$#-.H*)$./#$"#/*5'%&#2&###5'&# 2 ,ax bx c #>"-2&(#3"4&$5"(*2&4&#+"#I9/&-*+#('$5"--)'&$#$"#*$+&,-./#+"#<9$8(*)$*+#-.H*)$./J#

K.//)%2"#8"+)#-.(+&L#$"#+"#H*/.+#>"-5)-*2#3"4&$5"(*2&+#&#I9/&-*+J#

FG#M"(&# 0,a #2&--&+#3"4&$5"(*2*#

# 2 .ax bx c ax z #

# N&#+"#2.--*2#>J('J#

# 2 .ax bx c ax z #

# @+"'&-"#

# 2 2 22ax bx c ax a x z z #

# 22bx a x z z c #

#2

;2z c

xb z a

>-&%#$,.#,%&$5&+# J$# #

>.(#3"4&$5"(*2*+#$"#EFG#2&--&+#

# ( ) ,I r t dt #89# ( )r t #>.-.B&+#<9$8(*)$#-.H*)$./J#

M,%.('"2#5)#+"#2&--&%#$"(&#5)#+"#3"4&$5"()'&+#2 .ax bx c ax z #

:G#M"(&# 0,c #2&--&+#3"4&$5"(*2*0#

'"'(!.')/'#$/'#!,$)!!)

#O#

# 2 .ax bx c xz c #

@+"'&-"#

# 2 2 2 2ax bx c x z xz c c #

# 2 2ax b xz z c #

# 2( ) 2x z a b z c #

# 2

2b z cxz a

#>-&%#$,.#>.(+.%#H.8+)%2"# J$# #

6.(#3"4&$5"(*2*+#$"#EFG#2&--&+0#

# ( ) ,I r t dt #89# ( )r t #"('+"#<9$8(*)$#-.H*)$./J#

M,%.('"2#5)#+"#2&--&%$"(&#5)#+"#3"4&$5"()$*2#

# 2 .ax bx c xz c #

?G#M"(&# +-*$)2*# 89.5-.+*8# 8.# --"$%"# +"# $5-P('2&# -&./&# 1 2,x x # >-.#

$"(&# 21 2( )( )ax bx c a x x x x #.+"'&-"#2&--&+#3"4&$5"(*2*0#

# 21 2 1( )( ) ( )ax bx c a x x x x z x x #

# 2 21 2 1( )( ) ( )a x x x x z x x #

# 22 1( ) ( )a x x z x x #

# 2 22 1ax ax z x z x #

# 2 21 2( )z a x z x ax #

#21 22 ;z x ax

xz a

>-&%#$,.#H.8+)%2"# J$# #

6.(# 3"4&$5"(*2*+# $"# EFG# 2&--&+# ( ) ,r t dt # 89# ( )r t # "('+"# <9$8(*)$#-.H*)$./J#

K98&#3Q.+9.-#3"4&$5"(*2&+#&#I9/&-*+#+"#$%&'()'&$#*$+&,-./&+0#

!"#$%&'(!)!)*'+',#-'&)R#

<#!!24 1dx

x x xJ## # # # # # 5=## #

2

1 .3 9

x dxx x

!! ! !

55##2

.5 6dx

x x!

!"#$%&#'(!<##1&B"# 0,a #2&--&+#3"4&$5"(*2*#*#>.-"#*#I9/&-*+0#

# 24 1 2x x x z #

@+"'&-"L# 2 2 24 1 4 4x x x xz z #

# 24 1x xz z #

#2

2 1(1 4 ) 1 ; .1 4zx z z xz#

@+"'&-"L#2

2

2 22 .(1 4 )z z

dx dzz

#

7"4&$5"()%2"#$"#*$+&,-./*$#&#5'"$"#$"#<*//*2J#1&--&+0#

#

2 2

2 2

2 2 2 2 2

2 2 2 22(1 4 ) (1 4 )2

1 1 1 2 2 421 4 1 4 1 4 1 4

z z z zz zdz dz

z z z z z zzz z z z

#

#2

2 2 2

2 2 12 2 ln .(1 )(2 2) 1 1

z z dz zdz C

z z z z z#

;$+&,-./*# *# <9$5*+# >.-.B&+# *$+&,-./# -.H*)$./L# 2&# 3,%*5'%&$# &# +"# H*/*+#2&--&+0#

5=#!1&B"# 9,c #2&--&+#3"4&$5"(*2*#*#5P+"#*#I9/&-*+J#1&--&+0#

# 2 3 9 9 3.x x xz xz #

@+"'&-"#

# 2 2 23 9 6 9x x x z xz #

# 23 6x xz z #

'"'(!.')/'#$/'#!,$)!!)

#S#

# 2 3 6xz x z #

# 2( 1) 3(1 2 )x z z #

#2

2 2 2

1 2 13 ; 6 .1 (1 )

z z zx dx dz

z z#

6-.#

#22

2 22

2

3(1 2 ) 11 11 63(1 2 ) (1 )3 9 31

zx z zzdx dz

z zx x zz

#

# #2

2 2

6 42(1 )z z

dzz

#

6.(#3,%*5'%&(#2&--&+0#

#

55##1&B"# 2 5 6 ( 2)( 3)x x x x #3Q.+)%2"#3"4&$5"(*2*$#&#+-&+"#+"#I9/&-*+J#T&2*0#

# 2 5 6 ( 2).x x z x #

@+"'&-"L# 2 2 25 6 ( 2)( 3) ( 2) .x x x x z x #

KJ2J+'J#

# 2( 3) ( 2)x z x #

#2

22 2 2

3 3 2 2; ;2 1 (1 )

x z zz x dx dz

x z z#

1&--&+0#

#2 2 2 2

2 2 22

2 2

2 2(1 ) (1 )3 2 3 2 2 25 6 21 1

z zdx z zdz dz

z z zx x z zz z

!

!"#$%&'(!)!)*'+',#-'&)FU#

! ! ! ! ! ! 2

12 ln .1 1dz z

Cz z

!

)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

5:#!2

.1

dx

x x x! ! ! ! ! ! ! ! 5;#!

2.

1 1 2dx

x x! ! !

5<#!2

2

5 6 .5 6

x x xdx

x x x!! !

!

B*/2!,0)+2!&?!*%&'()24'1'!*)23*0%24'#

5#! ;$+&,-./*#*#<)-2"(#

#2

( ) ,mP xdx

ax bx c# # # # # # # # # # # # # # # EFG#

89# ( )mP x #"('+"#>)/*$)2#*#('8.//"(#%J#

;$+&,-./*#EFG#$%&'()'&+#(*>.(#<)-2"(0#

# 212 2

( ) ( )mm

P x dxdx P x ax bx c Kax bx c ax bx c

#

89# 1( )mP x #"('+"#>)/*$)2#*#('8.//"(#%VFL#&V#8)$(+.$+"J#

T)&<*H*&$+"+# &# >)/*$)2*+# 1( )mP x # 5'&# 8)$(+.$+.#&# H.8+)'&$# >"-2&(#2&+)5"(#("#8)&<*H*&$+&4&#+"#>.H.8+9.-J#

!"#$%&'()'&+#*$+&,-./*0#

56## 2 2 4 .x x dx #

!"#$%&#'(!

#2 2 2 4 2

2 2

2 2

4 4 444 4

x x x x xx x dx dx dx

x x#

# 3 2 2

2( ) 4 .

4dxAx Bx Cx D x Kx

#

'"'(!.')/'#$/'#!,$)!!)

#FF#

K98&#5&-*49.-#+"#5P#.$"+#&#Q.-.3*2*+#("#<9$5*+#2&--&+0#

#4 2

3 2 2

2 2

4 1[( ) 4] '4 4

x x Ax Bx Cx D x Kx x

#

4 2 3 22 2

2 2 2

4 ( )3 2 ) 44 4 4

x x Ax Bx Cx D x KAx Bx C x

x x x# 4 2 2 2 3 24 ( 2 )( 4) ( )x x Ax Bx C x Ax Bx Cx D x K #

6.(#+-.$(<)-2*2&4&#2&--&+0#1 1, 0, , 0, 2.4 2

A B C D K #

6-.$5.%L#>.(#3,%*5'%&(#2&--&+0#

#3

2 2 2 224 4 2 ln( 4) .4

x xx x dx x x x C #

)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

6=#!3

2

1 .2 5

x x dxx x

! ! ! ! ! ! 65#!3

2

2 5 .1

x xdx

x x! ! ! !

6#!;$+&,-./*#*#<)-2"(#

# # #2

.( )m

dx

x b ax bx c# # # # # # # # # # # # # E:G#

;$+&,-./*#E:G#3,%*5'&+#2&#.$"#+"#3"4&$5"(*2*+0#

#1.x bt#@+"'&-"# 2 .

dtdxt#

!"#$%&'()'&+#*$+&,-./*0#

57#!2 2

.( 1) 1

dx

x x x#

!"#$%&#'(!!

7"4&$5"()%2"# 2

11 ; .dtx dxt t

#1&--&+0#

!"#$%&'(!)!)*'+',#-'&)F:#

#2

2

22

2 1 11 1 1 1 21 1 1

dtdttI

t t tt t t

#

#2 2

2 2

.1 3 3 3 1 3 3 13

dt dt t dtt t t t

t t t

#

;$+&,-./*# $"# 4*%*2# 3,%*5'&+#2&# 3"4&$5"(*2*$# &# >.-"# E)(&# +"# 5P+"G# +"#I9/&-*+J#

6.(#3,%*5'%&(#2&--&+0#

#.)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

66#!3 2

.( 1) 1

dx

x x x# # # # 67## 3

2

.1 2 72

dx

x x x#

68##4 2

.1

dx

x x x!

$CDAEF$G$!$!B$HAFCI$JK$D!L$CMG$JK#

;$+&,-./*#*#<)-2"(0#

# ( ) ,m n px a bx dx #

89# , ,m n p # %.$"# $92-.# -.H*)$./# 5'&# ,a b R # B9'&+# *%&'()24! *!@*,')'%3*24*&!N*%0+*24J#

;$+&,-./*#*#2"(*>"-2"L#$"#2"$P-"#&/&2&$+.-&#29$5#+"#3,%*5'&+#4&+"2#$"#$%"-*$#$,.#+-*#-.(+&+#4*%9&(&0#

FG##M"(&#'#"('+"#$92"-#*#>/)+"#2&--&+#3"4&$5"(*2*0# ,sx t #89#(#"('+"#('92"<*('*#2"#*#4),"/#*#>"-Q.('8"+#>"-#&2"-9&(*+#&#$92-.4&# , .m n #

'"'(!.')/'#$/'#!,$)!!)

#F?#

:G##M"(&# p Z # E>-.# ),p Q # 5J2J+'J# $"(&#rps# 5'&#

1 ,mZ

n#

.+"'&-"#2&--&+#3"4&$5"(*2*# n sa bx t #E(#W#&2"-9&(*#*#'#W#("GJ#

?G#M"(&#'# "('+"# +'P&("# rp

s# 5'&#

1 ,m Zn

# 89-(&#1 ,mp Z

n#

2&--&+#3"4&$5"(*2* .n n sa bx x t #

!"#$%&'()'&$#*$+&,-./&+0#

58#!3.

(1 )dx

x x!! ! ! 59#!

23.

1

xdx

x! ! 5"#!

44.

1dx

x!

!"#$%&#'(!58##;$+&,-./*#*#5'"$"#('"$)'&+#$"#+-.%+"$#

#11

232 (1 ) .I x x dx #

X"-&%2"#(&#1 1; , 2.2 3

m n p #

1&B"# 2p #"('+"#$92"-#*#>/)+"L#3"4&$5"()%2"0#

# 6x t #E(&>(&#[2,3] 6). #

# 56 .dx t #

1&--&+0#

#11

6 6 2 5 2 2 532( ) (1 ( ) ) 6 3(1 ) 6I t t t dt t t t dt #

#

5 2

3 2 2 2 26 6(1 ) (1 )t dt t dt

t t t#

2&#3,%*5'%&$#&#+"#H*/*+#2&--&+0#

#

#

#

#

!"#$%&'(!)!)*'+',#-'&)FA#

59##;$+&,-./*$#&#5'"$"#&#>.-.B&(*2#$"#+-.%+"$0#

#2 13 2(1 ) .I x x #

X"-&%2"#(&#2 11; ; .3 2

m n p #

1&B"#1 3m

n# 2&--&+# 3"4&$5"(*2*# 2 231 ,x t # >-&%# $,.#

2 23 ( 1) .xdx t t dt #

T&2*0#1 5

2 2 2 2 2 32( ) 3 ( 1) 3 ( 1) 3 2 3 ;5tI t t t dt t dt t t C ##

89# 231 .t x #

5"##;$+&,-./*$#&#5'"$"#&#>.-.B&(*2#$"#+-.%+"$0#

#1

4 444

(1 ) .1dxI x dxx

!

X"-&%2"#(&#10; 4; .4

m n p #

X"-&%2"#(&#1 1

4m

Zn

#>)-#1 1 1 0 .

4 4m

p Zn

#

KJ2J+'J#3"4&$5"()%2"0# 4 4 41 .x x t #

KJ2J+'J#

# 4 4 4 44 44

1 11 .1 1

x t x x xt t

#

6.(#5&-*4*2*+#2&--&+0#

#3

4 54.

( 1)tdx dtt

#

7"4&$5"()%2"#$"#*$+&,-./*$#&#5'"$"J#1&--&+0#

'"'(!.')/'#$/'#!,$)!!)

#FC#

#1 4 3 24

4 444 54

( 1)(1 ) .

1( 1)

t t t dtx dx dtt tt

#

;$+&,-./*#*#<9$5*+#>.-.B&+#*$+&,-./#+"#$%"#<9$8(*)$*#-.H*)$./J#

6.(#3,%*5'%&(#2&--&+0#

4 44 4

44

1 1 1 1 1 1 1 1ln arctan ln arctan .4 1 2 4 21 1

t x xI t C Ct xx

#

)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

69#!2

3

2 1 .x x dxx

! ! ! 6"#! 2 23 ( 1) .x x x dx ! ! ! !

6:#!3 41 .x

dxx

! ! ! ! ! ! 6;#! 44 4 .x x dx ! !

6<#!4 5

.1dx

x x! ! ! ! ! ! 7=#! 33 3 .x x dx !

!

!

!

!"#!$%&'()*+*!*!,-%./*0%'1'!&)*(0%0+'&)*.'!!!!

!"##$%#&'#(%&'# (sin , cos ) ,R x x dx #)!"#*+#!#',-&'#.+/*,01/#23401/356#

7'#23,&0/#%#8'29(0&-,-':#:%22%&#;'<%/=',0:0>#

# tan .2x

t #?&'-%2'# 2

2arctan 2arctan , .

2 1x dt

t x t dxt#

# 22 2

sin 2 22sin .1sin cos

2 2

xtx

x x t#

#22 2 2

22 2 2 2

1 4 1cos . cos 1 sin 1 .

1 (1 ) 1t t t

x x xt t t

#

@+*%#;'<%/=',+32#/'#)!"#:%22%&>#

#2

2 2 2

2 1 2. .

1 1 1t t dt

Rt t t

#

A23B#0/&%92350#0#.+/*,01/%<%#&2091/1:%&20*%#,-/='221-%&#/'#0/&%9235#&'#.+/*,01/0&#23401/356#C"#7',%#.+/*,01/0#0/&%9235#851&',1/#*+,-&0/##

( sin , cos ) (sin , cos ),R x x R x x #823#/',%#.+/*,01/0#0/&%9235#',-&'#

&%*# ,083,# ,0/D# 3&'-%2'# :%22%&# ;'<%/=',0:0># 2cos ; sin 1 ;x t x t #

2

1arcsin ; .1

x t dx dtt

#

E"##7',%# (sin , cos ) (sin , cos ),R x x R x x # 823# /',%# .+/*,01/0#/'/0/&%9235#',-&'#&%*#,083,# cos ,x #:%22%&#;'<%/=',0:0>#

# 2

2sin ; arcsin ; ; cos 1 .

1

dtx t x t dx x tt

#

!"#$%&'(!)!)*'+',#-'&)C#

F"#7',%# ( sin , cos ) (sin , cos ),R x x R x x # 823# /',%# .+/*,01/0#/'/0/&%9235#',-&'#G0.&#,083,#sin , cos ,x x #:%22%&#;'<%/=',0:0>#

# 2tan ; arctan ; .1dt

x t x x dxt#

H%I'##

#

2

2 2 222

2 22 2 2 2

2 2

sinsin tancossin

sin cossin cos tan 1 1cos cos

xx x txx

x xx x x tx x

#

3&'-%2'##

#2

sin1

txt#

22

2 2

1cos 1 sin 1 .

1 1

tx x

t t#

J"# K/&%9235%&# %# &080&# sin cosm nx xdx # ,-/='221-%/# /'# 0/&%9235%# &'#.+/*,01/%<%#23401/35%#:%#*'&1#;'<%/=',0:%>#

# 0"# 7',%#"#',-&'#/+:'2#&%*B#;'<%/=',1(:'# cos .x t #

# 00"#7',%###',-&'#/+:'2#&%*B#;'<%/=',1(:'# sin .x t #

# 000"#7',%# ,m n #(3/'#/+:23#/3&L235#G0.&#;M3&1(:'#.12:+53&>#

# # # # # 2 21 cos 2 1 cos 2sin ; cos .2 2

x xx x #

#####0<"#7',%# ,m n #(3/'#/+:23#G0.&#:%#,-%/(3#&'#*+/='2&3B#################;'<%/=',1(:'# tan x t #1,%#ctan .x t #

N"# K/&%9235%&#%#&080&>#

# 0"# sin sin .mx nxdx #

# 00"## sin cos .mx nxdx #

# 000"# cos cos .mx nxdx #

'"'(!.')/'#$/'#!,$)!!)

#E#

O9(0=-%/# =+*%# ;M3&+32# .12:+3&B# &'# 4053&# 0# *%:0# 8'2:%/=+2# &%*#0/&%920:0#:%#:%&1='/#%#;'<%/=',0:0&6#

# 1sin sin (cos( ) cos( ) )2

mx nx m n x m n x #

# 1sin cos (sin( ) sin( ) )

2mx nx m n x m n x #

# 1cos cos (cos( ) cos( ) ).2

mx nx m n x m n x #

P'#/(%-,1-%/#0/&%9235%&#&2091/1:%&20*%#

2#!4

3

cos .sin

x dxx

! ! ! ! ! ! ! 3#!3

4

sin .cos

x dxx

! ! !

4## 3 4sin cos .x xdx ## # # # # 5#! 6ctan .xdx !

#!6(7*897:#!!

2#!O'<%/=',1(:'# 2

2cos ;sin 1 ;

1

dtx t x t dxt#

#4 4 4

3 2 3 2 22

cossin (1 ) (1 )1

x t dt tdx dtx t tt

!

! !

3#!#3 2 2

4 4 4

sin sin sin 1 cos sincos cos cos

x x x xdx dx xdxx x x

!

!2

4 3

cos1 1 1

sin .3cos cos

sin

x tt

xdx dt dt Ct x x

xdx dt!

4#!7'#*'&'#23,&# 3,m #;'<%/=',1(:'# cos ; sinx t xdx dt #

# 3 4 2 4 2 4sin cos (1 cos ) cos sin (1 )x xdx x x xdx t t dt #

!"#$%&'(!)!)*'+',#-'&)F#

#5 7 5 7cos cos .5 7 5 7t t x xC C #

5#!#6 4

6 46 4 2

cos cosctan cossin sin sin

x x dxxdx dx xx x x

!

!64 2

2 2 2

22

2

ctan

ctan cos .sin sin 1

cos1

x udx dx ux x du dux x u

uxu

!

!!"#$%&'()%'*+,#%-."''P'#/(%-,1-%/#0/&%9235%&>#

2#!6

5

sin .cos

x dxx

!! ! ! 3#!3

23

sin .cos

x dxx

! ! ! ! 4#!3 3

.sin cos

2 2

dxx x

!

5#! 2sin cos .x xdx ! ! ;#!4

6

sin .cos

x dxx

!

@+*%#0#;M3&+32#.12:+53&>## # # # # # # sin( ) sin(( ) ( ))x x #

# # # # # # # cos( ) cos(( ) ( ))x x #

/(%,1/0###

;## .sin( ) sin( )

dxx x

# # # <#! .sin sin

dxx a

!

/01-2,1&3$

1 sin(( ) ( ))sin( ) sin( ) sin( ) sin( ) sin( )

dx x x dxx x x x

#

# 1 sin( ) cos( ) sin( ) cos( )sin( ) sin( ) sin( )

x x x x dxx x

#

'"'(!.')/'#$/'#!,$)!!)

#J#

# 1 cos( ) cos( )sin( ) sin( ) sin( )

x xdx dxx x

#

# 1 sin( )ln , sin( ) 0.sin( ) sin( )

x Cx

#

<#!H%I'#cos cos2 2x a x a

a #3&'-%2'#*%:0>#

#cos

2 21sin sin 2cos sin cos

2 2

x a x adx

x a x ax a a#

#cos cos sin sin

1 2 2 2 22cos sin cos

2 2

x a x a x a x a

dxx a x aa

#

#cos sin1 2 2

2cos sin cos2 2

x a x a

dx dxx a x aa

#

#sin1 2ln , cos 0; sin sin .

cos cos2

x a

C a x ax aa

#

P'#/(%-,1-%&#0/&%9235%&>##

"## .sin cos 3

dxx x

! ! ! ! ! =#!2 2

4 4

sin cos .sin cos

x x dxx x

!

>#! 2 2

sin 3cos.

3sin 5cosx x

dxx x

!

/01-2,1&3$

O'<%/=',1(:'># tan ; arctan ; 2arctan2 2x xt t x t #

!"#$%&'(!)!)*'+',#-'&)N#

#2

2 2 2

2 2 1; sin ; cos .1 1 1dt t tdx x xt t t

#

=6:6&-6#

#2 2

2 2 2

2 2 2

21 12

2 1 2 1 3 3sin cos 3 31 1 1

dt dtdx t t

t t t t tx xt t t

#

# 2 22 .4 2 2 2 1

dt dtt t t t

#

=##O'<%/=',1(:'# 2 2 2

1tan ; ; sin ; cos .1 1 1

dt tx t dx x xt t t

#

?&'-%2'#

#

2

2 2 2 2

44 4 2

2 2 2 2

1sin cos 1 1

1sin cos 1(1 ) (1 )

tx x dtt tdx

tx x tt t

2

2 4 .( 1)( 1)

tdt

t t!

>#!! 2 2 2 2

sin 3cos sin3sin 5cos 3(1 cos ) 5cos

x x xdxdxx x x x

!

! 2 2

cos33sin 5(1 sin )

xdxx x 2 2

sin cos3

3 2cos 5 2sinxdx xdx

x x#

# 2 2

(cos ) (sin )3 .

3 2cos 5 2sind x d x

x x#

2?##P'#<'2&%&1-%&#,%#

# 1 2

sin cos( sin cos ) ( sin cos ) ( sin cos )n n n

dx A x B x dxCa x b x a x b x a x b x

#

*+# , ,A B C #Q#*1/,&3/&%6#

/01-2,1&3$

'"'(!.')/'#$/'#!,$)!!)

#R#

$%# &'# (%&'# .( sin cos )n n

dxIa x b x

#@+*%#8'2=12#:%&1='/#8324035%# &'#

0/&%920:0&#:%22%&>#

# 1 1

( cos sin ) cos sin( sin cos ) ( sin cos )n n n

d a x b x a x b xIa x b x a x b x

#

#2

2 1

( cos sin ) cos sin( 1)

( sin cos ) ( sin cos )n n

a x b x a x b xn

a x b x a x b x#

#2 2

2

( cos sin ) ( cos sin )( 1) .( sin cos )n

a x b x b x a xn dxa x b x

#

A23#

# 22 2 1

1 sin cos( 2)( 1)( ) ( sin cos )n n n

b x a xI n In a b a x b x

#

9('#I'#=+-%(#&2%9+326#'!"#$%&'()%'*+,#%-."''P'#/(%-,1-%/#0/&%9235%&>#

<#! .cos( ) cos( )

dxx a x b

!! ! ! ! "#! .sin( ) cos( )

dxx a x b

!! ! !

=#! .cos cos

dxx a

# # # # # # # >#! .sin sin

dxx a

! !

2?#! .cos cos

dxx a

# # # # # # # 22#! .sin 2cos 5

dxx x

#

23## .3sin cos 1

dxx x

## # # # # 24#!2

2 4

sin .sin cos

xdxx x

#

25#!2 4

4 4

sin cos.

cos sinx x

dxx x

## # # # # 2;## 2 2

sin 2cos.

3sin 7 cosx x

dxx x

#

2<## 3 .(2sin 5cos )dxx x

# # # # # 2"#! 2 2

3sin 4cos.

4sin 5cosx x

dxx x

!

!"#$%&'(!)!)*'+',#-'&)S#

2=#! 3 .(sin 3cos )dx

x x## # 2>#!

1

1

cos2 ,

sin2

n

n

x a

dxx a

# # #

3?##sin

2 , .sin

2

nx a

dx n Nx a

!

32#!T'2&%&1/0#,%#<5%/#

1 1

sin( cos ) ( cos ) ( cos )n n n

dx A x dxBa b x a b x a b x

#

# # # 2 , ( , , )( cos )ndxC A B C

a b xQ#*1/,&3/&%B# , (| | | | .n N a b #

!!!!!!!!!!!!!!

!"#!$%&'()*+*!*!,-%./*0%'1'!&)2%/3'%4'%&'!!!!!!"#$%&'()'&$#*$+&,-./&+0#

5#!! sin .xxe xdx ! ! ! ! ! 6#!3

.1 2

x

x

e dxe! ! ! 7#!

5 102

.1

x xx

dx

e e e!

8#! .1 1x x

dx

e e!! ! 9#!! ln( 1 1 )x x dx !

!"#$%&#'(!

5##12.+)%3"#4.5+*$#6"# sin .2

ix ixe exi

#7&--&+#

# (1 ) (1 )1sin ( )

2 2

ix ixx x x i x xe e

xe xdx xe dx x e e dxi i

!

! (1 ) (1 )2 3

1 1 12 1 (1 ) 1 (1 )

x i x ix xe e C

i i i i i!

! ( (sin cos ) cos ) .2

xe x x x x C !

6##1"8&$9"()%3"0# 3 3; ; .x x xe u e u e dx du #

#3 3 3 2

1 2 ( 1 2 ) (1 2 ) 2 1

x x x

x x x

e dx e e dx u du u due e e u u u

!

!21 1 1 1 1 1

ln | 2 1 | .2 4 4 2 1 4 4 4

uu du u u C

u!

7##1"8&$9"()%3"0# 10 10 101 ; 10 .10

x x x

e u e dx du e dx du #7&--&+0#

!"#$%&'(!)!)*'+',#-'&):#

#5 2 1

10 5 102

101

1x x xx

dx duu u u u

e e e e!

! 2 22 310 10

( 1)(1 ( 1))[ ( 1) ( 1du du

u u u u uu u u u#

8#!!( 1 1 )( 1 1 )

xx

xx x x

e ue dx duI

u u ue dx due e e#

# 2 2

1 1 1 1 1 1 .(1 1 ) 2 2u u u udu du du

u u u u u#

;.(#<,%*9'%&(#3*--&+0#

# 1 1 1 1ln .2 1 1

x x

x x

e eI C

e e#

9#!12.+)%3"#3&+)9"$#=.->*./&#+"#*$+&,-*3*+0#

ln( 1 1 )

1 1 1 1ln( 1 1 )2 1 1 ) 1 1

x x u

x x dx dx dux x x x

v x

!

1 1 1ln( 1 1 )

2 1 1 1 ) 1x

x x x dxx x x x

#

1 ( 1 1 )ln( 1 1 )2 2 1x x x dxx x x

x x#

1 ( 1 1 ) ln( 1 1 )2 2 1x x x dx x x x

x x#

1 1 1 11 14 1 4 1

x xdx dxx x

#

'"'(!.')/'#$/'#!,$)!!)

#?#

1 1 1ln( 1 1 )

2 4 1 1x x x

x x x dxx x

#

1ln( 1 1 ) arcsin , | | 1.

2 2x

x x x x C x #

)*+,-'./0-.12&+-$3*..!"#$%&'()'&$#*$+&,-./&+0#

5#! 2 cos .xx e xdx ! ! ! 6#! 1.1

x

x

ee

! ! ! ! ! 7#!3 62

.1

x xx

dx

e e e!

8#!!4

2 .1 4

x

x

e dxe

! ! ! ! 9#! 2 2ln ( 1 ) .x x dx ! :#!2 3

ln .(1 )

x dxx

#

;## 1arccos .x dxx

!! ! ! "#! 2arctan ln(1 ) .x x x dx !

!!!