X2 t04 02 trig integrals (2012)

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  • Trig Integrals

  • Trig Integrals(1) Standard Integrals

    caxaaxdx cos1sin

  • Trig Integrals(1) Standard Integrals

    caxaaxdx cos1sin caxaaxdx sin1cos

  • Trig Integrals(1) Standard Integrals

    caxaaxdx cos1sin caxaaxdx sin1cos caxaaxdx tan1sec2

  • Trig Integrals(1) Standard Integrals

    caxaaxdx cos1sin caxaaxdx sin1cos caxaaxdx tan1sec2 dxaxaxaxdx cossintan

  • Trig Integrals(1) Standard Integrals

    caxaaxdx cos1sin caxaaxdx sin1cos caxaaxdx tan1sec2 dxaxaxaxdx cossintan

    caxa

    ORcaxa

    seclog1 coslog1

  • xx nn cosor sin2

  • xx nn cosor sin2 xdxsin cx cos

  • xx nn cosor sin2 xdxsin xdx2sin

    cx cos

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cx cos

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin dxxx 2cos1sin

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin dxxx 2cos1sin

    xdxduxu

    sincos

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin dxxx 2cos1sin

    xdxduxu

    sincos

    duu21

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin dxxx 2cos1sin

    xdxduxu

    sincos

    duu21

    cxx

    cuu

    coscos3131

    3

    3

  • xx nn cosor sin2 xdxsin xdx2sin dxx2cos121

    cxx

    2sin21

    21

    cx cos

    xdx3sin xdxx 2sinsin dxxx 2cos1sin

    xdxduxu

    sincos

    duu21

    cxx

    cuu

    coscos3131

    3

    3 xu

    xxxx

    PowerOddsome power

    cosUsecos1sin Substitute

    sinsin as Factorise22

    2

  • xdx4sin

  • xdx4sin dxx 22sin

  • xdx4sin dxx 22sin dxx 22cos141

  • xdx4sin dxx 22sin dxx 22cos141 dxxx 2cos2cos2141 2

  • xdx4sin dxx 22sin dxx 22cos141 dxxx 2cos2cos2141 2

    dxxx 4cos1212cos2141

  • xdx4sin dxx 22sin dxx 22cos141 dxxx 2cos2cos2141 2

    dxxx 4cos1212cos2141

    dxxx 4cos212cos22341

  • xdx4sin dxx 22sin dxx 22cos141 dxxx 2cos2cos2141 2

    dxxx 4cos1212cos2141

    dxxx 4cos212cos22341cxxx

    4sin

    812sin

    23

    41

  • xdx4sin dxx 22sin dxx 22cos141 dxxx 2cos2cos2141 2

    dxxx 4cos1212cos2141

    dxxx 4cos212cos22341cxxx

    4sin

    812sin

    23

    41

    xx

    x

    PowerEvenpowersome

    2cos121sin Substitute

    sin as Factorise

    2

    2

  • xdx5sin

  • xdx5sin dxxx 22sinsin

  • xdx5sin dxxx 22sinsin dxxx 22cos1sin

  • xdx5sin dxxx 22sinsin dxxx 22cos1sin

    xdxduxu

    sincos

  • xdx5sin dxxx 22sinsin dxxx 22cos1sin

    xdxduxu

    sincos

    duuu

    duu42

    22

    21

    1

  • xdx5sin dxxx 22sinsin dxxx 22cos1sin

    xdxduxu

    sincos

    duuu

    duu42

    22

    21

    1

    cxxx

    cuuu

    53

    53

    cos51cos

    32cos

    51

    32

  • xx nn cos and sin3

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g.

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g. xdxxx sincos1cos 25

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g.xdxdu

    xusin

    cos

    xdxxx sincos1cos 25

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g.xdxdu

    xusin

    cos

    xdxxx sincos1cos 25

    duuu

    duuu57

    25 1

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g.xdxdu

    xusin

    cos

    xdxxx sincos1cos 25

    duuu

    duuu57

    25 1

    cxx

    cuu

    68

    68

    cos61cos

    81

    61

    81

  • xx nn cos and sin3Usually done by substitution xuxu cosor sin

    xdxxi 35 sincos e.g.xdxdu

    xusin

    cos

    xdxxx sincos1cos 25

    duuu

    duuu57

    25 1

    cxx

    cuu

    68

    68

    cos61cos

    81

    61

    81

    powerhigher eUsually th aseither Choose

    u

    oddpowersBoth

  • xdxxii 36 cossin

  • xdxxii 36 cossin xdxxx cossin1sin 26

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin dxxx 22 sin1sin

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin dxxx 22 sin1sin dxxx 42 sinsin

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin dxxx 22 sin1sin dxxx 42 sinsincxxxxx 4sin

    3212sin

    41

    832sin

    41

    21

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin dxxx 22 sin1sin dxxx 42 sinsincxxxxx 4sin

    3212sin

    41

    832sin

    41

    21

    cxx 4sin321

    81

  • xdxxii 36 cossin xdxxx cossin1sin 26

    xdxduxu

    cossin

    duuu

    duuu86

    26 1

    cxx

    cuu

    97

    97

    sin91sin

    71

    91

    71

    uevenpoweroneoddpowerOne

    aseven Choose&

    xdxxiii 22 cossin dxxx 22 sin1sin dxxx 42 sinsincxxxxx 4sin

    3212sin

    41

    832sin

    41

    21

    cxx 4sin321

    81xx

    xxevenpowersBoth

    22

    22

    sin1cosor cos1sin Use

  • xx nn cotor tan4

  • xx nn cotor tan4 xdxtan cx coslog

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan xdx3tan

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan xdx3tan

    xdxxdxx

    dxxx

    tansectan

    1sectan2

    2

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan xdx3tan

    xdxxdxx

    dxxx

    tansectan

    1sectan2

    2

    xdxduxu

    2sectan

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan xdx3tan

    xdxxdxx

    dxxx

    tansectan

    1sectan2

    2

    xdxduxu

    2sectan

    xdxudu tan

  • xx nn cotor tan4 xdxtan cx coslog xdx2tan dxx 1sec2

    cxx tan xdx3tan

    xdxxdxx

    dxxx

    tansectan

    1sectan2

    2

    xdxduxu

    2sectan

    xdxudu tan

    cxx

    cxu

    coslogtan21

    coslog21

    2

    2

  • xdx4tan

  • xdx4tan

    xdxxdxx

    dxxx222

    22

    tansectan

    1sectan

  • xdx4tan

    xdxxdxx

    dxxx222

    22

    tansectan

    1sectan

    xdxduxu

    2sectan

  • xdx4tan

    xdxxdxx

    dxxx222

    22

    tansectan

    1sectan

    xdxduxu

    2sec