Integral for logarithm

26
| . กกกก

description

Integral for logarithm

Transcript of Integral for logarithm

Page 1: Integral for logarithm

| ��.���� �� ������

ก�������� ��ก�����ก��� ��ก�����

Page 2: Integral for logarithm

Calculus

2

�����ก��ก�ก�� ������ก��� ������ 2

ก�� ������ก������ก��������� ก!��������ก�������"�#$���%&��'(����ก������ก���)* ��- ก��............................................................................... ���................................�����............... ---------------------------------------------------------------------------------------------------------------------- ��� a , c ��� n ���� !� "�� #$%ก�%&� !��'��'ก%(�)*"ก+�(������,ก-��("���.��(/0+����)*"ก+�(���ก�%'�12 #$%5 ∫ du

u

1 = culn + #$%6 ∫ dua u = c

aln

a u+ �2� a> 0 ��� a ≠ 1

#$%7 ∫ dueu = ceu +

-�&%(:;<�=+���=#!>�%#� ∫ dx)x(f

1)x(g , ∫ ⋅ dxa)x(g )x(f ��� ∫ ⋅ dxe)x(g )x(f

������ !� u = f(x) ��� dx = u

du′

�">�;<�=+���D $(D�#ก�'��'�ก%$<�E2!2�$(D��% x �%�กF��= ���D�%�

�2�%�&� !��'��'ก%(�;G=>�� #$% 5 , 6 ��� 7 EG� G("$(D�=!�"$!�E���, <"&� !��'��'ก%(�$!�E���,

1. <"&� !���" ∫ ⋅ dxex242x2

D'0��-� >�� #$% ∫ dueu = ceu +

>&� u = 2x2 <�EG� u ′ = x4

∴ dx = u

du′

= x4

du

G("�(,� ∫ ⋅ dxex242x2

= ∫ ⋅x4

duex24 u

= ∫ ⋅ due6 u

= ce6 u +⋅

= ce62x2 +⋅ #

2. <"&� !���" ∫ ⋅ dx2)x3sin(12 )x3cos(

D'0��-� >�� #$% ∫ dua u = caln

a u+

>&� u = )x3cos(

<�EG� u ′ = )x3sin(3−

∴ dx = u

du′

= )x3sin()3(

du

G("�(,� ∫ ⋅ dx2)x3sin(12 )x3cos(

= ∫ −⋅

)x3sin()3(

du2)x3sin(12 u

= ∫ ⋅− du24 u

= c2ln

24 u+

⋅−

= c2ln

24 )x3cos(+

⋅− #

3. <"&� !���" ∫ dxx

e x

D'0��-� >�� #$% ∫ dueu = ceu +

>&� u = x

<�EG� u ′ = x2

1

Page 3: Integral for logarithm

Calculus

2

∴ dx = u

du′

=

x2

1du

G("�(,� ∫ dxx

e x

= ∫x2

1du

x

eu

= ∫ ⋅ due2 u

= ce2 u +⋅

= ce2 x + #

4. <"&� !���" ∫−

dx3x2

x242

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = 3x2 2 − <�EG� u ′ = x4

∴ dx = u

du′

= x4

du

G("�(,� ∫−

dx3x2

x242

= ∫ x4

du

u

x24

= ∫ duu

6

= culn6 +

= c3x2ln6 2 +− #

5. <"&� !���" ∫−

dxe32

e24x4

x4

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = x4e32 −

<�EG� u ′ = x4e)12(−

∴ dx = u

du′

= x4e)12(

du

G("�(,� ∫−

dxe32

e24x4

x4

= ∫− x4

x4

e)12(

du

u

e24

= ∫−

duu

2

= culn2 +−

= ce32ln2 x4 +−− #

6. <"&� !���" ∫ −dx

)x5tan(3

)x5(sec30 2

D'0��-� >�� #$% ∫ dua u = caln

a u+

>�� #$% <"&� !���" ∫ duu

1 = culn +

>&� u = )x5tan(3− <�EG� u ′ = )x5(sec)5( 2−

∴ dx = u

du′

= )x5(sec)5(

du2−

G("�(,� ∫ −dx

)x5tan(3

)x5(sec30 2

= ∫− )x5(sec)5(

du

u

)x5(sec302

2

= ∫−

duu

6

= culn6 +−

= c)x5tan(3ln6 +−− #

Page 4: Integral for logarithm

Calculus

3

7. <"&� !���" ∫ dxx

3 )x2ln(

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = ln(2x)

<�EG� u ′ = x2

2 = x

1

∴ dx = u

du′

= x1

du

G("�(,� ∫ dxx

3 )x2ln(

= ∫x

1du

x

3u

= ∫ du3u

= c3ln

3u+

= c3ln

3 )x2ln(+ #

�กKL+ก�%>&� ���� �::MNก�(กO�ก�%&� !��'��'ก%(� �� 1.2 (����� 3 ����) ก-�&�G u , &� !� u′������ !� dx �#ก$��" 2:#%K+ EG� 1.5 ���� ก-�&�G u , &� !� u′������ !� dx �#ก:�" !D� EG� 0.5 ���� &� !��'��'ก%(��#ก$��" 2:#%K+ EG� 1 ���� &� !��'��'ก%(��#ก:�" !D� EG� 0.5 ���� ��� !���" u >�.�ก�%�'��'�ก%$�#ก$��" EG� 0.5 ���� E2!��� !���" u >�.�ก�%�'��'�ก%$�#ก$��" EG� 0 ����

�����ก��ก�ก�� ������ก��� ������ 2.1

1. ∫ ⋅ dx5x123x2

D'0��-� >�� #$% ∫ dua u = caln

a u+ �2� a = 5

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ ⋅ dx5x123x2

=

=

=

= c55ln

4 3x + #

2. ∫ ⋅ dx2)x4sin(28 )x4cos(

D'0��-� >�� #$% ∫ dua u = caln

a u+ �2� a = 2

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ ⋅ dx2)x4sin(28 )x4cos(

=

=

Page 5: Integral for logarithm

Calculus

2

=

= c22ln

7 )x4cos( +⋅− #

3. ∫−

dx4x3

x122

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

dx4x3

x122

=

=

=

= c4x3ln2 2 +− #

4. ∫ dx7)x(

1)x2ln(

D'0��-� ���"<�ก )x2ln(7)x(

1 = x

7 )x2ln(−

>�� #$% ∫ dua u = caln

a u+ �2� a = 7

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ dx7)x(

1)x2ln(

=

=

=

= c7ln7

1)x2ln(

+ #

5. ∫+−

−dx

5x4x

12x62

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫+−

−dx

5x4x

12x62

=

=

=

= c5x4xln3 2 ++− #

6. ∫ +−⋅− dxe)x412( )1x62x(

D'0��-� >�� #$% ∫ dueu = ceu +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ +−⋅− dxe)x412( )1x62x(

=

=

Page 6: Integral for logarithm

Calculus

3

=

= ce2 )1x62x( +− +− #

7. ∫−+

−dx

xx61

30x102

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−+

−dx

xx61

30x102

=

=

=

= cxx61ln5 2 +−+− #

8. ∫−

dxx

e8 )x3(

D'0��-� >�� #$% ∫ dueu = ceu +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

dxx

e8 )x3(

=

=

=

= ce16 )x3( +− − #

9. ∫−

−dx

xx4

2x2

D'0��-� >�� #$% ∫ duu

1 = culn +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

−dx

xx4

2x2

=

=

=

= cxx4ln2

1 2 +−− #

10. ∫+

dx3e4

e32x2

x2

D'0��-� >�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

Page 7: Integral for logarithm

Calculus

4

G("�(,� ∫+

dx3e4

e32x2

x2

=

=

=

= c3e4ln4 x2 ++ #

ก�'������ก����ก���� ������ก���

>�:�" %(,"<-�����$��"�%(:)*"ก+�(�ก!��ก�%&� !��'��'ก%(� ;G=ก�%G1"$(D%!D2&%�ก�%$(,"&�%G("$(D�=!�"$!�E���, ก%K�ก�%�%(:)*"ก+�(�;G=ก�%$(,"&�%2(ก����)*"ก+�(�$%%ก=�

• ���)*"ก+�(� f(x) ���g(x) ����)*"ก+�(�/&���2 �%�=ก )x(g

)x(f D!�����)*"ก+�(�$%%ก=�

• �%�=ก)*"ก+�(� )x(g

)x(f D!������SO�ก'� �2������,ก-��("��" x >� f 2�กกD!�&%���!�ก(: �����,ก-��("��" x >� g

• ก�%�'��'�ก%$)*"ก+�(��������SO�ก'� <��%(:)*"ก+�(�G�D=ก�%$(,"&�% ���D�-�.�&�%����SOT1"�����SO !D����E�&� !��'��'�ก%$

• ��ก<�ก)*"ก+�(�$%%ก=����D=("2�)*"ก+�(����=#!>�%#��SO !D����2�/<�+ Ua &%� Ue >�:�"ก%K���<$��"$(,"&�%ก!��ก�%&� !��'��'�ก%$ G("$(D�=!�"$!�E���,

$(D�=!�"

1. <"&� !���" ∫−

dxxx2

36

D'0��-� ���"<�ก xx2

36

− =

)1x2(x

36

>�� #$% ∫ duu

1 = culn +

>&� u = )1x2( −

<�EG� u ′ = x2

2 = x

1

∴ dx = u

du′

=

x

1du

G("�(,� ∫−

dxxx2

36

= ∫−

dx)1x2(x

36

= ∫x

1du

)u(x

36

= ∫ du)u(

36

= culn36 +

= c1x2ln36 +− #

2. <"&� !���" ∫−

dxx2

x3x122

2

D'0��-� ���"<�ก 2

2

x2

x3x12 − = x

1

2

36 ⋅−

Page 8: Integral for logarithm

Calculus

8

G("�(,� ∫−

dxx2

x3x122

2

= ∫ ⋅− dx)x1

23

6(

= cxln23

x6 +−

&%� ���"<�ก 2

2

x2

x3x12 − = x23

6 −

>�� #$% ∫ duu

1 = culn +

>&� u = )x2( <�EG� u ′ = 2

∴ dx = u

du′

= 2

du

G("�(,� ∫−

dxx2

x3x122

2

= ∫ − dx)x2

36(

= ∫ ∫− dxx2

3dx6

= 2

du

u

3x6 ∫−

= duu

1

2

3x6 ∫−

= culn2

3x6 +−

= cx2ln2

3x6 +− #

3. <"&� !���" ∫ −

+dx

1x2

1x6

D'0��-� ���"<�ก 1x2

1x6

+ = 1x2

43

−+

>�� #$% ∫ duu

1 = culn +

>&� u = )1x2( − <�EG� u ′ = 2

∴ dx = u

du′

= 2

du

G("�(,� ∫ −

+dx

1x2

1x6

= ∫ −+ dx)

1x2

43(

= ∫ −+ dx)

1x2

43(

= ∫ ∫ −+ dx

1x2

4dx3

= ∫+ 2

du

u

4x3

= cduu

2x3 ++ ∫

= culn2x3 ++

= c1x2ln2x3 +−+ #

4. <"&� !���" ∫+

+dx

1x

x2x42

3

D'0��-� ���"<�ก 1x

x2x42

3

+

+ = 1x

x2x4

2 +−

>�� #$% ∫ duu

1 = culn +

>&� u = )1x( 2 + <�EG� u ′ = x2

∴ dx = u

du′

= x2

du

G("�(,� ∫+

+dx

1x

x2x42

3

= ∫+

− dx)1x

x2x4(

2

= ∫ ∫+

− dx1x

x2xdx4

2

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

4

3x6

3

1x61x2

+−

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

x2

x4x4

x4

x2x41x

3

32

+

++

Page 9: Integral for logarithm

Calculus

9

= ∫− x2du

ux2

2x4 2

= ∫− duu1

x2 2

= culnx2 2 +−

= c1xlnx2 22 ++− #

5. <"&� !���" ∫−

−dx

1e

3ex4

x4

D'0��-� ���"<�ก 1e

3ex4

x4

− = 1e

e23

x4

x4

−−

>�� #$% ∫ duu

1 = culn +

>&� u = )1e( x4 − <�EG� u ′ = x4e4

∴ dx = u

du′

= x4e4

du

G("�(,� ∫−

−dx

1e

3ex4

x4

= ∫−

− dx1e

e23

x4

x4

= ∫ ∫−

− dx1e

e2dx3

x4

x4

= ∫− x4

x4

e4

du

u

e2x3

= ∫− duu

1

2

1x3

= culn2

1x3 +−

= c1eln2

1x3 x4 +−− #

�����ก��ก�ก�� ������ก��� ������ 2.2

1. dxxx3

12∫

D'0��-� ���"<�ก xx3 − = )1x3(x − >�� #$% >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� dxxx3

12∫

=

=

=

= #

2. dxx2x

82∫−

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

x4

x4

x4x4

e2

e33

3

e3e1

+−

+−+−

Page 10: Integral for logarithm

Calculus

8

D'0��-� ���"<�ก x2x 2 − = )x

21(x 2 −

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� dxx2x

82∫−

=

=

=

= #

3. ∫ +

−dx

1x2

5x6

D'0��-� ���"<�ก

1x2

5x6

+

− = 1x2

83

+−

>�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ +

−dx

1x2

5x6

=

=

=

= #

4. ∫ −

+−dx

1x2

7x8x4 2

D'0��-� ���"<�ก

1x2

7x8x4 2

+−

= 1x2

43x2

−+−

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ −

+−dx

1x2

7x8x4 2

=

=

=

= #

5. ∫+

−dx

2e5

4e2x3

x3

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

3x6

3

5x61x2

+

−+

Page 11: Integral for logarithm

Calculus

2

D'0��-� ���"<�ก 2e5

4e2x3

x3

+

− =

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫+

−dx

2e5

4e2x3

x3

=

=

=

= #

!-�������ก��ก�ก�� ������ก��� 2.1

1. c55ln

4 3x +

2. c22ln

7 )x4cos( +⋅−

3. c4x3ln2 2 +−

4. c7ln7

1)x4ln(

+

5. c5x4xln3 2 ++−

6. ce2 )1x62x( +− +−

7. cxx61ln5 2 +−+−

8. ce16 )x3( +− −

9. cxx4ln2

1 2 +−−

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

4

3x6

7x6

x2x4

3x2

7x8x41x2

2

2

+−

+−

+−−

�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2!G("��,

x3x3 e24e52 +−+

Page 12: Integral for logarithm

Calculus

2

10. c3e4ln4 x2 ++

!-�������ก��ก�ก�� ������ก��� 2.2

1. c1x3ln8 +−

2. cx

21ln4 +−

3. c1x2ln4x3 ++−

4. c1x2ln2x3x 2 +−+−

5. c2e5ln5

4x2 x3 +++−

29. ∫+−

++−dx

5x4x

8x21x20x52

23

D'0��-� ���"<�ก 5x4x

8x21x20x52

23

+−

++− =

&� ∫ dx ;G=>�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

8x21x20x55x4x 232 ++−+−

Page 13: Integral for logarithm

Calculus

3

G("�(,� ∫+−

++−dx

5x4x

8x21x20x52

23 =

=

=

=

31. ∫−

−dx

3e7

1e6x4

x4

D'0��-� ���"<�ก 3e7

1e6x4

x4

− =

&� ∫ dx ;G=>�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

−dx

3e7

1e6x4

x4 =

=

=

=

�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2! G("��,

x4x4 e61e73 +−+−

Page 14: Integral for logarithm

Calculus

4

11. ∫ dxx

e123

2x

2

D'0��-� >�� #$% ∫ dueu = ceu +

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ dxx

e123

2x

2

=

=

=

= ce32x

2

+− #

12. ∫−

dxx43

e8 x43

D'0��-� >�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

dxx

e8 )x3(

=

=

=

Page 15: Integral for logarithm

Calculus

5

= #

13. ∫ −dx

)x4sin(23

)x4cos(48

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ −dx

)x4sin(23

)x4cos(48

=

=

=

= #

14. ∫ ⋅ dxee12x3ex3

D'0��-� >�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ ⋅ dxee12x3ex3

=

=

=

Page 16: Integral for logarithm

Calculus

6

= #

15. ∫ −dx

1)x3tan(4

)x3(sec30 2

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ −dx

1)x3tan(4

)x3(sec30 2

=

=

=

= #

15. ∫ −− dx)ee( 2x3x3

D'0��-� ���"<�ก 2x3x3 )ee( −− =

=

>&� u = ���>&� v =

<�EG� u′= ��� v′ =

∴ dx = u

du′

= du

Page 17: Integral for logarithm

Calculus

7

∴ dx = vdv′

= dv

G("�(,� ∫ −− dx)ee( 2x3x3

=

=

=

=

= #

16. ∫ − dxe)ee( xxx2

D'0��-� ���"<�ก xxx2 e)ee( − =

=

>&� u = ���>&� v =

<�EG� u′= ��� v′ =

∴ dx = udu′

= du

∴ dx = vdv′

= dv

G("�(,� ∫ − dxe)ee( xxx2

=

=

=

=

= #

17. ∫ −dx

1)x3tan(4

)x3(sec30 2

Page 18: Integral for logarithm

Calculus

8

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ −dx

1)x3tan(4

)x3(sec30 2

=

=

=

= #

18. ∫ dxe)x2cos()x2sin(x6 )2x2(2cos22

D'0��-� >�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ dxe)x2cos()x2sin(x6 )2x2(2cos22

=

=

=

= #

19. ∫+

dxe

8et2

t4

D'0��-� ���"<�ก t2

t4

e

8e + =

Page 19: Integral for logarithm

Calculus

9

=

>&� u = ���>&� v =

<�EG� u′= ��� v′ =

∴ dx = udu′

= du

∴ dx = vdv′

= dv

G("�(,� ∫+

dxe

8et2

t4

=

=

=

=

= #

20. ∫−

dx)xsec(7

)xtan()xsec(x304

443

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

dx)xsec(7

)xtan()xsec(x304

443

=

=

=

= #

Page 20: Integral for logarithm

Calculus

10

21. ∫ dxe)x2(eccosx12 )3x2cot(322

D'0��-� >�� #$%

>&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ dxe)x2(eccosx12 )3x2cot(322

=

=

=

= #

22. ∫+−

−dx

e

6x4

)1x32x(

D'0��-� ���"<�ก )1x32x(e

6x4

+−

− =

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫+−

−dx

e

6x4

)1x32x(

=

=

=

=

Page 21: Integral for logarithm

Calculus

11

= #

23. ∫ dx)x5ln(x

202

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ dx)x5ln(x

202

=

=

=

= #

24. ∫−

dx)ecos(36

)esin(e24x2

x2x2

D'0��-� >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

dx)ecos(36

)esin(e24x2

x2x2

=

=

=

= #

Page 22: Integral for logarithm

Calculus

8

ก�%�%(:)*"ก+�(�ก!��&� !��'��'ก%(� >�:�" %(,"<-�����$��"�%(:)*"ก+�(�ก!��ก�%&� !��'��'ก%(� ;G=ก�%G1"$(D%!D2&%�ก�%$(,"&�%G("$(D�=!�"$!�E���, ก%K�ก�%�%(:)*"ก+�(�;G=ก�%$(,"&�%2(ก����)*"ก+�(�$%%ก=�

• ���)*"ก+�(� f(x) ���g(x) ����)*"ก+�(�/&���2 �%�=ก )x(g

)x(f D!�����)*"ก+�(�$%%ก=�

• �%�=ก)*"ก+�(� )x(g

)x(f D!������SO�ก'� �2������,ก-��("��" x >� f 2�กกD!�&%���!�ก(: �����,ก-��("��" x >� g

• ก�%�'��'�ก%$)*"ก+�(��������SO�ก'� <��%(:)*"ก+�(�G�D=ก�%$(,"&�% ���D�-�.�&�%����SOT1"�����SO !D����E�&� !��'��'�ก%$

• ��ก<�ก)*"ก+�(�$%%ก=����D=("2�)*"ก+�(����=#!>�%#��SO !D����2�/<�+ Ua &%� Ue >�:�"ก%K���<$��"$(,"&�%ก!��ก�%&� !��'��'�ก%$ G("$(D�=!�"$!�E���,

<"&� !��'��'ก%(�$!�E���,

25. dxxx3

12∫

D'0��-� ���"<�ก xx3 − = )1x3(x − >�� #$% >�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� dxxx3

12∫

− =

=

=

=

26. dxx2x

82∫−

D'0��-� ���"<�ก x2x 2 − = )x

21(x 2 −

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� dxx2x

82∫−

=

Page 23: Integral for logarithm

Calculus

8

=

=

=

27. ∫ +

−dx

1x2

5x6

D'0��-� ���"<�ก 1x2

5x6

+

− = 1x2

83

+−

>�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫ +

−dx

1x2

5x6 =

=

=

=

28. ∫ −

+−dx

1x2

7x8x4 2

D'0��-� ���"<�ก 1x2

7x8x4 2

+− = 1x2

43x2

−+−

&� ∫ −dx

1x2

4 >�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

4

3x6

7x6

x2x4

3x2

7x8x41x2

2

2

+−

+−

+−−

&� ∫ +dx

1x2

8

;G=>�� #$% ∫ += cUlndUU

1

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

Page 24: Integral for logarithm

Calculus

9

G("�(,� ∫ −

+−dx

1x2

7x8x4 2 =

=

=

=

29. ∫+−

++−dx

5x4x

8x21x20x52

23

D'0��-� ���"<�ก 5x4x

8x21x20x52

23

+−

++− =

&� ∫ dx ;G=>�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫+−

++−dx

5x4x

8x21x20x52

23 =

=

=

=

30. ∫+

−dx

2e5

4e2x3

x3

D'0��-� ���"<�ก 2e5

4e2x3

x3

+

− =

&� ∫ dx >�� #$%

>�� #$% >&� u = <�EG� u ′ =

$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%

8x21x20x55x4x 232 ++−+−

�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2! G("��,

x3x3 e24e52 +−+

Page 25: Integral for logarithm

Calculus

10

∴ dx = u

du′

= du

G("�(,� ∫+

−dx

2e5

4e2x3

x3 =

=

=

=

31. ∫−

−dx

3e7

1e6x4

x4

D'0��-� ���"<�ก 3e7

1e6x4

x4

− =

&� ∫ dx ;G=>�� #$%

>�� #$% >&� u = <�EG� u ′ =

∴ dx = u

du′

= du

G("�(,� ∫−

−dx

3e7

1e6x4

x4 =

=

=

=

-�$�:�::MNก�(กO�ก�%&� !��'��'ก%(� 2

1. c55ln

4 3x + 2. c22ln

7 )x4cos( +⋅−

�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2! G("��,

x4x4 e61e73 +−+−

Page 26: Integral for logarithm

Calculus

11

3. c4x3ln2 2 +− 4. c7ln7

1)x4ln(

+

5. c5x4xln3 2 ++− 6. ce2 )1x62x( +− +−

7. cxx61ln5 2 +−+− 8. ce16 )x3( +− −

9. cxx4ln2

1 2 +−− 10. c3e4ln4 x2 ++

11. ce32x

2

+− 12. ce4 x43 +− −

13. c)x4sin(23ln6 +−− 14. ce4x3e +

15. c1)x3tan(4ln2

5+− c

e6

1x2e

6

1x6

x6 +−− 16. ce2

1e

3

1 x2x3 +−

17. c1)x3tan(4ln2

5+− 18. ce

4

3 )2x2(2cos +

19. ce

4e

2

1t2

t2 +− 20. c)xsec(7ln2

15 4 +−−

21. ce2 )3x2cot( +− 22. ce

2

)1x32x(+−

+−

23. c)x5ln(ln10 2 + 24. c)ecos(36ln4 x2 +−

25. c1x3ln8 +− 26. cx

21ln4 +−

27. c1x2ln4x3 ++− 28. c1x2ln2x3x 2 +−+−

29. c5x4xln2x2

5 22 ++−− 30. c2e5ln5

4x2 x3 +++−

31. ce12

11x

3

1 x4 ++