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    Method of Integration Word

    Uploaded by

    Anis Farhanah
    1. The document discusses methods of integration using techniques like substitution and integration by parts. 2. Several integrals involving exponential, logarithmic, trigonometric and other functions are presented. 3. The methods shown produce intermediate expressions that are then further simplified through additional integrations or substitutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd

    Method of Integration Word

    Uploaded by

    Anis Farhanah
    26 views8 pages
    1. The document discusses methods of integration using techniques like substitution and integration by parts. 2. Several integrals involving exponential, logarithmic, trigonometric and other functions are presented. 3. The methods shown produce intermediate expressions that are then further simplified through additional integrations or substitutions.

    Original Description:

    Advance Calculus

    Copyright

    © Attribution Non-Commercial (BY-NC)

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    DOCX, PDF, TXT or read online from Scribd

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    1. The document discusses methods of integration using techniques like substitution and integration by parts. 2. Several integrals involving exponential, logarithmic, trigonometric and other functions are presented. 3. The methods shown produce intermediate expressions that are then further simplified through additional integrations or substitutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    26 views8 pages

    Method of Integration Word

    Uploaded by

    Anis Farhanah
    1. The document discusses methods of integration using techniques like substitution and integration by parts. 2. Several integrals involving exponential, logarithmic, trigonometric and other functions are presented. 3. The methods shown produce intermediate expressions that are then further simplified through additional integrations or substitutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 8

    Method of Integration

    Recall

    Consider

    =
    =

    which produces a more difficult expression


    Now consider

    =

    =

    Consider

    V U
    (
    (
    (
    ]
    d U V U V
    (
    (
    (
    ]
    d
    x x e
    x

    (
    (
    (
    ]
    d x e
    x
    ( )
    x ( )
    (
    (
    (
    ]
    d
    x e
    x
    ( )
    d
    x
    2
    2
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    e
    x
    ( )
    x
    x
    2
    2
    |

    \
    |
    |
    .
    d e
    x
    ( )

    (
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    e
    x
    ( )
    x
    x
    2
    2
    |

    \
    |
    |
    .
    e
    x

    (
    (
    (
    (
    ]
    d
    x x e
    x

    (
    (
    (
    ]
    d x x ( ) d e
    x
    ( ) ( )

    (
    (
    (
    ]
    d
    x e
    x
    x e
    x
    ( )
    d x ( )
    (
    (
    (
    ]
    d x e
    x
    x e
    x
    ( )
    1 ( )
    (
    (
    (
    ]
    d
    x e
    x
    x e
    x
    (
    (
    (
    ]
    d x e
    x
    e
    x
    k +
    x x
    2
    e
    x

    (
    (
    (
    ]
    d x e
    x
    ( )
    x
    2
    ( )

    (
    (
    (
    ]
    d
    =
    =

    which produces a more difficult expression

    Now consider



    =

    =



    Consider
    =
    =

    Consider

    x e
    x
    ( )
    d
    x
    3
    3
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    3
    3
    |

    \
    |
    |
    .
    e
    x
    ( )
    x
    x
    3
    3
    |

    \
    |
    |
    .
    d e
    x
    ( )

    (
    (
    (
    (
    ]
    d
    x
    3
    3
    |

    \
    |
    |
    .
    e
    x
    ( )
    x
    x
    3
    3
    |

    \
    |
    |
    .
    e
    x

    (
    (
    (
    (
    ]
    d
    x x
    2
    e
    x

    (
    (
    (
    ]
    d x x
    2
    ( )
    d e
    x
    ( ) ( )

    (
    (
    (
    ]
    d x
    2
    e
    x
    x e
    x
    ( )
    d x
    2
    ( )

    (
    (
    (
    ]
    d
    x
    2
    e
    x
    x e
    x
    ( )
    2 x ( )
    (
    (
    (
    ]
    d x
    2
    e
    x
    2 x x e
    x

    (
    (
    (
    ]
    d
    |

    \
    |
    |
    |
    .

    x
    2
    e
    x
    2 e
    x
    x 1 ( )

    e
    x
    x
    2
    2 x 2 +
    ( )
    k +
    x x e
    x

    (
    (
    (
    ]
    d x x d e
    x

    ( )

    (
    (
    (
    ]
    d
    x e
    x
    x e
    x

    ( )
    d x ( )
    (
    (
    (
    ]
    d x e
    x
    x e
    x
    1 ( )
    (
    (
    (
    ]
    d +
    x e
    x
    x e
    x
    (
    (
    (
    ]
    d + x e
    x
    e
    x

    x x
    2
    e
    x

    (
    (
    (
    ]
    d x x
    2
    d e
    x

    ( )

    (
    (
    (
    ]
    d
    =

    =


    Consider
    =
    =
    =



    consider
    =
    =

    =



    x
    2
    e
    x
    x e
    x

    ( )
    d x
    2
    ( )

    (
    (
    (
    ]
    d x
    2
    e
    x
    2 x x e
    x

    (
    (
    (
    ]
    d +
    x
    2
    e
    x
    2 x e
    x
    e
    x

    ( )
    +
    x ln x ( )
    (
    (
    (
    ]
    d x ln x ( ) ( ) 1 ( )
    (
    (
    (
    ]
    d
    x ln x ( ) ( ) d x ( ) ( )
    (
    (
    (
    ]
    d xln x ( ) x x d ln x ( ) ( )
    (
    (
    (
    ]
    d
    x ln x ( ) x x ( )
    1
    x
    |

    \
    |
    |
    .

    (
    (
    (
    ]
    d x ln x ( ) x 1
    (
    (
    (
    ]
    d
    x ln x ( ) x k +
    x x ln x ( ) ( )
    (
    (
    (
    ]
    d x ln x ( ) ( ) d
    x
    2
    2
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    ln x ( ) x
    x
    2
    2
    |

    \
    |
    |
    .
    d ln x ( ) ( )
    (
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    ln x ( ) x
    x
    2
    2
    |

    \
    |
    |
    .
    1
    x
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    ln x ( ) x
    x
    2
    |

    \
    |
    |
    .
    (
    (
    (
    ]
    d
    x
    2
    2
    |

    \
    |
    |
    .
    ln x ( )
    x
    2
    4
    k +
    consider

    =

    =

    =


    consider

    =
    =
    =


    consider
    x x
    3
    ln x ( ) ( )
    (
    (
    (
    ]
    d x ln x ( ) ( ) x
    3
    ( )

    (
    (
    (
    ]
    d
    x ln x ( ) ( ) d
    x
    4
    4
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    4
    ln x ( )
    4
    x
    x
    4
    4
    d ln x ( ) ( )
    (
    (
    (
    (
    ]
    d
    x
    4
    ln x ( )
    4
    x
    x
    4
    4
    |

    \
    |
    |
    .
    1
    x
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    x
    4
    ln x ( )
    4
    x
    x
    3
    4
    |

    \
    |
    |
    .
    (
    (
    (
    (
    ]
    d
    x
    4
    ln x ( )
    4
    x
    4
    16
    k +
    x x sin 2 x ( ) ( )
    (
    (
    (
    ]
    d x x d
    cos 2 x ( )
    2
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    ]
    d
    x cos 2 x ( )
    2
    x
    cos 2 x ( )
    2
    |

    \
    |
    |
    .
    d x ( )
    (
    (
    (
    ]
    d
    x cos 2 x ( )
    2
    x
    cos 2 x ( )
    2
    |

    \
    |
    |
    .
    1 ( )
    (
    (
    (
    ]
    d +
    x cos 2x ( )
    2
    x
    cos 2x ( )
    2
    |

    \
    |
    |
    .
    (
    (
    (
    ]
    d +
    xcos 2x ( )
    2
    sin 2x ( )
    4
    + k +
    x x cos 2x ( ) ( )
    (
    (
    (
    ]
    d x x d
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    ]
    d
    =
    =

    =

    =


    consider


    =

    =

    =

    =



    x sin 2x ( )
    2
    x
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    d x ( ) ( )
    (
    (
    (
    ]
    d
    x sin 2 x ( )
    2
    x
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    1 ( )
    (
    (
    (
    ]
    d
    x sin x ( )
    2
    1
    2
    x sin 2x ( ) ( )
    (
    (
    (
    ]
    d

    (
    (
    (


    x cos 2x ( )
    2
    x
    cos 2x ( )
    2
    |

    \
    |
    |
    .
    (
    (
    (
    ]
    d +
    xcos 2x ( )
    2
    sin 2x ( )
    4
    + k +
    x x cos 2x ( ) ( )
    (
    (
    (
    ]
    d x x d
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    ]
    d
    x sin 2x ( )
    2
    x
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    d x ( ) ( )
    (
    (
    (
    ]
    d
    x sin 2 x ( )
    2
    x
    sin 2x ( )
    2
    |

    \
    |
    |
    .
    1 ( )
    (
    (
    (
    ]
    d
    x sin 2x ( )
    2
    1
    2
    x sin 2x ( ) ( )
    (
    (
    (
    ]
    d

    (
    (
    (


    x sin 2x ( )
    2
    1
    2
    cos 2x ( )
    2
    |

    \
    |
    |
    .

    x sin 2x ( )
    2
    cos 2x ( )
    4
    + k +
    consider



    =

    =

    =

    =


    =

    consider



    =

    =

    x x
    2
    cos x ( ) ( )
    (
    (
    (
    ]
    d x x
    2
    d sin x ( ) ( ) ( )
    (
    (
    (
    ]
    d
    x
    2
    ( )
    sin x ( ) x sin x ( ) ( ) d x
    2
    ( ) ( )

    (
    (
    (
    ]
    d
    x
    2
    ( )
    sin x ( ) x 2 x ( )sin x ( )
    (
    (
    (
    ]
    d
    x
    2
    ( )
    sin x ( ) 2 x x sin x ( ) ( )
    (
    (
    (
    ]
    d

    (
    (
    (


    x
    2
    ( )
    sin x ( ) 2 x cos x ( ) sin x ( ) + ( )
    x
    2
    ( )
    sin x ( ) 2 x cos x ( ) + 2 sin x ( )
    t e
    st
    t ( )
    (
    (
    (
    ]
    d t t d
    e
    st

    s
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    t ( )
    e
    st

    s
    |

    \
    |
    |
    .
    t
    e
    st

    s
    |

    \
    |
    |
    .
    d t ( ) ( )
    (
    (
    (
    (
    ]
    d
    t e
    st

    s
    t
    e
    st
    s
    |

    \
    |
    |
    .
    1 ( )
    (
    (
    (
    (
    ]
    d +
    =

    =

    consider

    =

    =

    =

    =

    =
    =
    t e
    st

    s
    t
    e
    st
    s
    |

    \
    |
    |
    .
    (
    (
    (
    (
    ]
    d +
    t e
    st

    s
    e
    st
    s
    2

    t e
    st
    sin kt ( ) ( )
    (
    (
    (
    ]
    d t sin kt ( ) ( ) ( ) d
    e
    st

    s
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d
    sin kt ( ) ( )
    e
    st

    s
    |

    \
    |
    |
    .
    t
    e
    st

    s
    |

    \
    |
    |
    .
    d sin kt ( ) ( ) ( )
    (
    (
    (
    (
    ]
    d
    sin kt ( ) ( ) e
    st

    s
    1
    s
    t e
    st
    ( )
    k cos kt ( ) ( )
    (
    (
    (
    ]
    d +
    sin kt ( ) ( ) e
    st

    s
    k
    s
    t cos kt ( ) ( ) d
    e
    st

    s
    |

    \
    |
    |
    .
    |

    \
    |
    |
    .

    (
    (
    (
    (
    ]
    d +
    sin kt ( ) ( ) e
    st

    s
    k
    s
    e
    st

    s
    |

    \
    |
    |
    .
    cos kt ( ) ( ) t
    e
    st

    s
    |

    \
    |
    |
    .
    d cos kt ( ) ( ) ( )
    (
    (
    (
    (
    ]
    d

    (
    (
    (
    (

    +
    sin kt ( ) ( ) e
    st

    s
    k e
    st

    s
    2
    cos kt ( ) ( )
    k
    s
    |

    \
    |
    |
    .
    t
    e
    st
    s
    |

    \
    |
    |
    .
    k sin kt ( ) ( )
    (
    (
    (
    (
    ]
    d +
    sin kt ( ) ( ) e
    st

    s
    k e
    st

    s
    2
    cos kt ( ) ( )
    k
    2
    s
    2
    t e
    st
    ( )
    sin kt ( ) ( )
    (
    (
    (
    ]
    d

    Hence


    1
    k
    2
    s
    2
    +
    |

    \
    |
    |
    |
    .
    t e
    st
    ( )
    sin kt ( ) ( )
    (
    (
    (
    ]
    d
    sin kt ( ) ( ) e
    st

    s
    k e
    st

    s
    2
    cos kt ( ) ( )
    t e
    st
    ( )
    sin kt ( ) ( )
    (
    (
    (
    ]
    d
    k
    2
    s
    2
    +
    k
    2
    |

    \
    |
    |
    |
    .
    sin kt ( ) ( ) e
    st

    s
    k e
    st

    s
    2
    cos kt ( ) ( )

    (
    (
    (

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