integral trigonometry
DESCRIPTION
Integral Trigonometry functionTRANSCRIPT
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Calculus
2
�����ก��ก�ก�� ������ก��� ������ 3
ก�� ������ก������ก��������ก !��� ����-�ก...............................................................................�� �.............................................� ���............... -------------------------------------------------------------------------------------------------------------------------------- ��� c ������������ �� u �������ก�� � ��! "��# x �%!#ก�#&�����'��'ก# ���ก�� �!#�(ก)*'!' *�+ ���,
�%!#8 ∫ duusin = cucos +− �%!#9 ∫ duucos = cusin +
�%!#10 ∫ duusec2 = cutan + �%!#11 ∫ duueccos 2 = cucot +−
�%!#12 ∫ ⋅ duutanusec = cusec + �%!#13 ∫ ⋅ duucotueccos = cueccos +−
�%!#14 ∫ duutan = cusecln +
= cucosln +−
�%!#15 ∫ duucot = cusinln +
�%!#16 ∫ duusec = cutanusecln ++ �%!#17 ∫ duueccos = cucotueccosln +−
�!"#��� cosec u ��;� �<�=+����� csc u �����ก�� �!#�(ก)*'!' <กก>� ����� Asin2 � �<�=+����� 2)A(sin ��� v �������ก�� � ��! "��# x ก@��*�#��# AB���%!#�&����,=+�B��>�����+�<"ก � ! "�<��� ;�&�����'��'ก# !��=���,
1. ;�&���� �� ∫ − dx)x23sin(x24 2
"'C��>� B���%!# ∫ duusin = cucos +−
B&� u = )x23( 2− ;�=+� u′ = )x4(−
������ dx = u
du′
= )x4(
du
−
+ �� ,� ∫ − dx)x23sin(x24 2
= ∫ − )x4(
du)usin(x24
= ∫− du)usin(6
= c)ucos(6 +−−
= c)x23cos(6 2 +− # 2. ;�&���� �� ∫ + dx)e25cos(e16 x4x4
"'C��>� ∫ duucos = cusin +
B&� u = )e25( x4+ ;�=+� u′ = )e8( x4
������ dx = u
du′
= )e8(
dux4
+ �� ,� ∫ + dx)e25cos(e16 x4x4
= ∫)e8(
du)ucos(e16
x4x4
= ∫ du)ucos(2
= c)usin(2 +
= c)e25sin(2 x4 ++ #
3. ;�&���� �� ∫−
−dx
1x2
1x2tan4
"'C��>� ∫ duutan = cusecln +
B&� u = 1x2 − = 2
1
)1x2( −
;�=+� u′ = 1x22
2
− =
1x2
1
−
Calculus
3
������ dx = u
du′
=
1x2
1du
−
+ �� ,� ∫−
−dx
1x2
1x2tan4
= ∫−
−1x2
1du
1x2
)utan(4
= ∫ du)utan(4
= c)usec(ln4 +
= c1x2secln4 +− #
&#�����B���%!# ∫ duutan = cucosln +−
;�=+� ∫−
−dx
1x2
1x2tan4 = c1x2cosln4 +−− #
4. ;�&���� �� ∫ ⋅ dx)x
4tan()
x
4sec(
x
202
"'C��>� ∫ ⋅ duutanusec = cusec +
B&� u = )x4
( = )x4( 1−
;�=+� u′ = 2x4 −− = 2x
)4(−
������ dx = u
du′
= 2x
)4(du−
+ �� ,� ∫ ⋅ dx)x
4tan()
x
4sec(
x
202
= ∫ −⋅
2
2
x
)4(du
)utan()usec(x
20
= ∫ ⋅− du)utan()usec(5
= c)usec(5 +−
= c)x
4sec(5 +− #
5. ;�&���� �� ∫ dx)2(sec)2(12 x32x3
"'C��>� B���%!# ∫ duusec2 = cutan +
B&� u = )2( x3 ;�=+� u′ = 2ln)2(3 x3
������ dx = u
du′
= 2ln)2(3
dux3
+ �� ,� ∫ dx)2(sec)2(12 x32x3
= ∫2ln)2(3
du)u(sec)2(12
x32x3
= ∫ du)u(sec2ln
4 2
= c)utan(2ln
4+
= c)2tan(2ln
4 x3 + #
Calculus
4
�����ก��ก�ก�� ������ก��� 3.1
;�&�����'��'ก# !��=���, 1. ;�&���� �� ∫ dx)x2cos(x24 43
"'C��>� B���%!# B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ dx)x2cos(x24 43
=
=
=
= #
2. ;�&���� �� ∫ ⋅ dx)e(eccose4 x2x
"'C��>� B���%!# B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ ⋅ dx)e(eccose4 x2x
=
=
=
= #
3. ;�&���� �� ∫ ⋅ dxxtanxsecx
12
"'C��>� B���%!#
B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ ⋅ dxxtanxsecx
12
=
=
=
= #
4. ;�&���� �� ∫ dx))x2cot(ln(x
20
"'C��>� B���%!#
B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ dx))x2cot(ln(x
20
=
=
=
= #
∫ duusec = cutanusecln ++
5. ;�&���� �� ∫ −− dx)xx4sec()8x4( 2
Calculus
5
"'C��>� B���%!# B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ −− dx)xx4sec()8x4( 2
=
=
=
= #
B�A���# ,��#�=*���*�#�&�����'��'ก# �����ก�� �!#�(ก)*'!'=+�� ��� ก�#�>���ก กD)� �����ก�� �!#�*'!'*���"<�# A���ก�� �ก���ก�#&�����'��'ก# ;E����� ,�!������>�� F B�ก#)����! "�%ก�'��'�ก#!�������ก�� �!#�(ก)*'!'*�&�<G;�� ก@��*�#�&�����'��'ก# �<ก�!��G;��=+�(+<B���%!#
∫ ± dx)wv( = ∫ ∫± wdxvdx
��&�����'��'ก# (+<ก�#������+�"<! "��# u &#�� v &#�����ก�� ����� H =+� ! "�<���ก�#�>���ก กD)� �����ก�� �!#�(ก)*'!'*��# A���ก�� �ก���ก�#&�����'��'ก# *�+ ���,
6. ;�&���� �� ∫−
−dx
)x4xtan(
8x42
"'C��>� ������;�ก=*�*��%!#&���� ∫ duutan
1
;�ก��ก กD)� AcotAtan
1=
;�=+�)x4xtan(
8x42 −
− = )x4xcot()2x(4 2 −−
B���%!# ∫ duucot = cusinln +
(+<B&� u = )x4x( 2 − ;�=+� u′ = 4x2 − = )2x(2 −
������ dx = u
du′
= )2x(2
du
−
+ �� ,� ∫−
−dx
)x4xtan(
8x42
= ∫ −− dx)x4xcot()2x(4 2
= ∫ −−
)2x(2
du)ucot()2x(4
= ∫ du)ucot(2
= c)usin(ln2 +
= c)x4xsin(ln2 2 +− #
��ก กD)����ก�� �!#�(ก)*'!'����>�� F *�+ ���,
1. Asin = ecAcos
1 2. Acos = Asec
1
3. Atan = Acot
1 4. Acot = Atan
1
Atan = Acos
Asin Acot = Asin
Acos
5. Asec = Acos
1 6. ecAcos = Asin
1
7. 1AcosAsin 22 =+ 8. AsecAtan1 22 =+ 9. AeccosAcot1 22 =+
10. )A2cos1(2
1Asin2 −=
11. )A2cos1(2
1Acos2 +=
12. AcosAsin2A2sin = \ 13. AsinAcosA2cos 22 −= A2cos = Asin21 2− = 1Acos2 2 − 14. BcosAsin2 = )BAsin()BAsin( −++ 15. BsinAsin2 = )BAcos()BAcos( +−− 16. BcosAcos2 = )BAcos()BAcos( −++
Calculus
6
7. ;�&���� �� ∫ dx)x6(tan24 2
"'C��>� ������;�ก=*�*��%!#&���� ∫ duutan2
;�ก��ก กD)� Atan2 = 1Asec2 − + �� ,� )x6(tan24 2 = )1)x6((sec24 2 − = 24)x6(sec24 2 − B���%!# ∫ duusec2 = cutan +
(+<B&� u = )x6( ;�=+� u′ = 6
������ dx = u
du′
= 6
du
+ �� ,� ∫ dx)x6(tan24 2
= ∫ − dx)24)x6(sec24( 2
= ∫∫ − dx24dx)x6(sec24( 2
= ∫∫ − dx246
du)u(sec24 2
= ∫∫ − dx24du)u(sec4 2
= cx24)utan(4 +−
= cx24)x6tan(4 +− #
8. ;�&���� �� ∫ dx)x2sin(
x102
"'C��>� ������;�ก=*�*��%!#&���� ∫ duusin
1 ;�ก
��ก กD)� ecAcosAsin
1=
+ �� ,� )x2sin(
x102
= )x2(eccosx10 2
B���%!# ∫ duueccos = cucotueccosln +−
B&� u = )x2( 2 ;�=+� u′ = )x4(
������ dx = u
du′
= x4
du
+ �� ,� ∫ dx)x2sin(
x102
= ∫ dx)x2(eccosx10 2
= ∫ x4
du)u(eccosx10
= ∫ du)u(eccos2
5
= c)ucot()u(eccosln2
5+−
= c)x2cot()x2(eccosln2
5 22 +− #
9. ;�&���� �� ( )∫
−dx
)x6cos(
12)x6sin(x52
2
"'C��>� ;�ก��ก กD)�
AtanAcos
Asin= �� ecAcos
Acos
1= ;�=+�
( ))x6cos(
12)x6sin(x52
2 − = )x6cos(
x60
)x6cos(
)x6sin(x522
2−
= )x6(eccosx60)x6tan(x5 22 −
B���%!# ∫ duutan = cusecln +
���%!# ∫ duusec = cutanusecln ++
(+<B&� u = )x6( 2 ;�=+� u′ = )x12(
������ dx = u
du′
= )x12(
du
+ �� ,� ( )∫
−dx
)x6cos(
12)x6sin(x52
2
= dx))x6sec(x60)x6tan(x5( 22 −∫
= ∫∫ − dx)x6sec(x60dx)x6tan(x5 22
= ∫∫ −)x12(
du)usec(x60
)x12(
du)utan(x5
= ∫∫ − du)usec(5du)utan(12
5
= c)utan()usec(ln5)usec(ln12
5++−
= c)x6tan()x6sec(ln5)x6sec(ln12
5 222 ++− #
B�A���# ,�! "�%ก�'��'�ก#!�<%�B�#%����J AJ���
ก�#���;����ก�%!#B�ก�#&�����'��'ก# �����ก u ����%ก!����&*���*=+�� ,�;�!���*��"�*#�A��A(+<B��& ก�>�� F��� �*��������� u ������<� dxB&�����
Calculus
7
udu′
��" ! "�%ก�'��'�ก#!;�=*��#�กL! "��#�+'*��ก
��B��=+�ก A�%!#������ก="�(+<=*�*����ก�� �!#�(ก)*'!'&#�����ก�� �B+�#�กL�<%� ! "�<�������
10. ;�&� ∫ ⋅⋅ dx)esin(e)x2(sec24 )x2tan()x2tan(2
"'C��>� ���B���%!# ∫ duusec2
(+<B&� u = )x2( ;�=+� u′ = 2
������ dx = u
du′
= 2
du
+ �� ,� ∫ ⋅⋅ dx)esin(e)x2(sec24 )x2tan()x2tan(2
= ∫ ⋅⋅2
du)esin(e)u(sec24 )utan()utan(2
= ∫ ⋅⋅ du)esin(e)u(sec12 )utan()utan(2
;��&@�"��! "�%ก�'��ก#!��ก;�ก*� )u(sec2 ��" < �*� )esin(e utanutan ⋅ ��+�"��B���%!# ∫ udusec2 =*�=+�
���B���%!# ∫ dueu
(+<B&� u = ))x2(tan( ;�=+� u′ = )x2(sec2 2
������ dx = u
du′
= usec2
du2
������ dx = u
du′
= usec2
du2
+ �� ,� ∫ ⋅⋅ dx)esin(e)x2(sec24 )x2tan()x2tan(2
= ∫ ⋅⋅usec2
du)esin(e)x2(sec24
2)u()u(2
;��&@�=*���*�#��# A! "��# x B&�&*+=�=+� ��ก;�ก��,< �*����ก�� ���������! "�%)�<%� ;E�=*���*�#�B���%!# ∫ dueu =+�
���B���%!# ∫ duusin
(+<B&� u = )e( )x2tan( ;�=+� u′ = )x2tan(2 e)x2(sec2
������ dx = u
du′
= )x2tan(2 e)x2(sec2
du
+ �� ,� ∫ ⋅⋅ dx)esin(e)x2(sec24 )x2tan()x2tan(2
= ∫⋅
⋅⋅)x2tan(2
)x2tan(2
e)x2(sec2
du)usin(e)x2(sec24
= ∫ du)usin(12
= c)ucos(12 +−
= c)ecos(12 )x2tan( +− #
�����ก��ก�ก�� ������ก��� 3.2
;�&�����'�'ก# !��=���,
1. ;�&���� �� ( )∫ − dx)x2tan(4)x2cos(x18 32
"'C��>� ������;�ก ( )∫ − dx)x2tan(4)x2cos(x18 32
= &���� ∫ dx)x2cos(x18 32
(+<B���%!# (+<B&� u = ;�=+� u′ =
������ dx = u
du′
= du
&���� ∫ dx)x2tan(4
(+<B���%!# (+<B&� v = ;�=+� v′ =
������ dx = v
dv′
= du
+ �� ,� ( )∫ − dx)x2tan(4)x2cos(x18 32
Calculus
8
= =
=
=
= #
2. ∫−
−−dx
)xxtan(
)xxcsc()3x6(2
2
"'C��>� ;�ก��ก กD)�
+ �� ,� )xxtan(
)xxcsc()3x6(2
2
−
−−
= B���%!# (+<B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫−
−−dx
)xxtan(
)xxcsc()3x6(2
2
=
=
=
=
= #
3. ∫ dx))x2(ln(cotx
24 2
"'C��>� ������;�ก Acot 2 = 1Acsc2 − + �� ,� ))x2(ln(cot 2 =
;�=+� ))x2(ln(cotx
24 2 =
&� ∫ dx))x2(ln(cscx
24 2
(+<B���%!# B&� u =
;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ dx))x2(ln(cotx
24 2
=
=
=
= #
4. ∫ dx)x4(sin24 2
"'C��>� ������;�ก + �� ,� )x4(sin 2 = = B���%!# ∫ duucos = cusin +
B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫ dx)x4(sin24 2
=
=
=
=
=
=
= #
5. ;�&���� �� ∫−
dx)x4sec(
)x4tan(8)x4csc(12
"'C��>� ������;�ก )x4sec(
)x4tan(8)x4csc(12 −
Calculus
9
= )x4sec(
)x4tan(8
)x4sec(
)x4csc(12−
= = B���%!# ∫ duucot ���%!# ∫ duusin
B&� u = ;�=+� u′ =
������ dx = u
du′
= du
+ �� ,� ∫−
dx)x4sec(
)x4tan(8)x4csc(12
=
=
=
=
=
= #