integral trigonometry

9
| . กกกก

Upload: arporn-intachai

Post on 23-Mar-2016

215 views

Category:

Documents


1 download

DESCRIPTION

Integral Trigonometry function

TRANSCRIPT

Page 1: Integral Trigonometry

| ��.���� �� ������

ก�������� ��ก�����ก��� ���ก����

Page 2: Integral Trigonometry

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

Page 3: Integral Trigonometry

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 + #

Page 4: Integral Trigonometry

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

Page 5: Integral Trigonometry

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( −++

Page 6: Integral Trigonometry

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&�����

Page 7: Integral Trigonometry

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

Page 8: Integral Trigonometry

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 −

Page 9: Integral Trigonometry

Calculus

9

= )x4sec(

)x4tan(8

)x4sec(

)x4csc(12−

= = B���%!# ∫ duucot ���%!# ∫ duusin

B&� u = ;�=+� u′ =

������ dx = u

du′

= du

+ �� ,� ∫−

dx)x4sec(

)x4tan(8)x4csc(12

=

=

=

=

=

= #