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Integral for logarithm
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| ��.���� �� ������
ก�������� ��ก�����ก��� ��ก�����
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
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 +−− #
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��-� >�� #$% ∫ dua u = caln
a u+ �2� a = 2
>&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ ⋅ dx2)x4sin(28 )x4cos(
=
=
Calculus
2
=
= c22ln
7 )x4cos( +⋅− #
3. ∫−
dx4x3
x122
D'0��-� >�� #$% ∫ 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��-� >�� #$% ∫ dueu = ceu +
>&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ +−⋅− dxe)x412( )1x62x(
=
=
Calculus
3
=
= ce2 )1x62x( +− +− #
7. ∫−+
−dx
xx61
30x102
D'0��-� >�� #$% ∫ 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��-� >�� #$%
>&� u = <�EG� u ′ =
∴ dx = u
du′
= du
Calculus
4
G("�(,� ∫+
dx3e4
e32x2
x2
=
=
=
= c3e4ln4 x2 ++ #
ก�'������ก����ก���� ������ก���
>�:�" %(,"<-�����$��"�%(:)*"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 ⋅−
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�%$(,"&�%
x2
x4x4
x4
x2x41x
3
32
−
+
++
Calculus
9
= ∫− x2du
ux2
2x4 2
= ∫− duu1
x2 2
= culnx2 2 +−
= c1xlnx2 22 ++− #
5. <"&� !���" ∫−
−dx
1e
3ex4
x4
D'0��-� ���"<�ก 1e
3ex4
x4
−
− = 1e
e23
x4
x4
−−
>�� #$% ∫ 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�%$(,"&�%
x4
x4
x4x4
e2
e33
3
e3e1
−
+−
+−+−
Calculus
8
D'0��-� ���"<�ก x2x 2 − = )x
21(x 2 −
>�� #$% >&� 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�%$(,"&�%
3x6
3
5x61x2
+
−+
Calculus
2
D'0��-� ���"<�ก 2e5
4e2x3
x3
+
− =
>�� #$% >&� 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 +−+
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�%$(,"&�%
8x21x20x55x4x 232 ++−+−
Calculus
3
G("�(,� ∫+−
++−dx
5x4x
8x21x20x52
23 =
=
=
=
31. ∫−
−dx
3e7
1e6x4
x4
D'0��-� ���"<�ก 3e7
1e6x4
x4
−
− =
&� ∫ dx ;G=>�� #$%
>�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫−
−dx
3e7
1e6x4
x4 =
=
=
=
�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2! G("��,
x4x4 e61e73 +−+−
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(
=
=
=
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
=
=
=
Calculus
6
= #
15. ∫ −dx
1)x3tan(4
)x3(sec30 2
D'0��-� >�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ −dx
1)x3tan(4
)x3(sec30 2
=
=
=
= #
15. ∫ −− dx)ee( 2x3x3
D'0��-� ���"<�ก 2x3x3 )ee( −− =
=
>&� u = ���>&� v =
<�EG� u′= ��� v′ =
∴ dx = u
du′
= du
Calculus
7
∴ dx = vdv′
= dv
G("�(,� ∫ −− dx)ee( 2x3x3
=
=
=
=
= #
16. ∫ − dxe)ee( xxx2
D'0��-� ���"<�ก xxx2 e)ee( − =
=
>&� u = ���>&� v =
<�EG� u′= ��� v′ =
∴ dx = udu′
= du
∴ dx = vdv′
= dv
G("�(,� ∫ − dxe)ee( xxx2
=
=
=
=
= #
17. ∫ −dx
1)x3tan(4
)x3(sec30 2
Calculus
8
D'0��-� >�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ −dx
1)x3tan(4
)x3(sec30 2
=
=
=
= #
18. ∫ dxe)x2cos()x2sin(x6 )2x2(2cos22
D'0��-� >�� #$%
>&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ dxe)x2cos()x2sin(x6 )2x2(2cos22
=
=
=
= #
19. ∫+
dxe
8et2
t4
D'0��-� ���"<�ก t2
t4
e
8e + =
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=
>&� u = ���>&� v =
<�EG� u′= ��� v′ =
∴ dx = udu′
= du
∴ dx = vdv′
= dv
G("�(,� ∫+
dxe
8et2
t4
=
=
=
=
= #
20. ∫−
dx)xsec(7
)xtan()xsec(x304
443
D'0��-� >�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫−
dx)xsec(7
)xtan()xsec(x304
443
=
=
=
= #
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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(
=
=
=
=
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= #
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
=
=
=
= #
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25. dxxx3
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−
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 −
>�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� dxx2x
82∫−
=
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=
=
=
27. ∫ +
−dx
1x2
5x6
D'0��-� ���"<�ก 1x2
5x6
+
− = 1x2
83
+−
>�� #$%
>�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫ +
−dx
1x2
5x6 =
=
=
=
28. ∫ −
+−dx
1x2
7x8x4 2
D'0��-� ���"<�ก 1x2
7x8x4 2
−
+− = 1x2
43x2
−+−
&� ∫ −dx
1x2
4 >�� #$%
>�� #$% >&� 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
>�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
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G("�(,� ∫ −
+−dx
1x2
7x8x4 2 =
=
=
=
29. ∫+−
++−dx
5x4x
8x21x20x52
23
D'0��-� ���"<�ก 5x4x
8x21x20x52
23
+−
++− =
&� ∫ dx ;G=>�� #$%
>�� #$% >&� u = <�EG� u ′ =
∴ dx = u
du′
= du
G("�(,� ∫+−
++−dx
5x4x
8x21x20x52
23 =
=
=
=
30. ∫+
−dx
2e5
4e2x3
x3
D'0��-� ���"<�ก 2e5
4e2x3
x3
+
− =
&� ∫ dx >�� #$%
>�� #$% >&� u = <�EG� u ′ =
$(D�#ก�'��'�ก%$�����SO !D��ก'� �%(:)*"ก+�(�;G=ก�%$(,"&�%
8x21x20x55x4x 232 ++−+−
�� �' ก�%$(,"&�%��";<�=+�%��U���, >&��%�="�-�G(:/<�+>&2! G("��,
x3x3 e24e52 +−+
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∴ 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 =
=
=
=
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7 )x4cos( +⋅−
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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 ++