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Math Help - integrals yielding inverse trigonometric functions

  1. #1
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    integrals yielding inverse trigonometric functions

    Need Help badly with this:

    the answer should be

    according to back of book

    Thanks alot!
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  2. #2
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    Basically use the substitution t= ln x
    And the identity for inverse tangent functions.
    Shown below.
    Attached Thumbnails Attached Thumbnails integrals yielding inverse trigonometric functions-picture23.gif  
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  3. #3
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    Hello, ^_^Engineer_Adam^_^!

    I don't agree with the book either . . .



    The answer should be: according to back of book

    . . . . . . . . . . . .1 . . . . dx
    We have: .∫ ----------------
    . . . . . . . . .1 + (ln x) . x

    . . . . . . . . . - . . . . . . dx
    Let u = ln x . . du = ---
    . . . . . . . . . . . - . - . . .x

    . . . . . . . . . . . du
    Substitute: .∫ -------- . = . arctan(u)
    . . . . . . . . . . 1 + u

    . . . . - . . . . - . . . - . . . . . . .|e
    Back-substitute: .arctan(ln x) |
    . . . . - . . . . - . . . - . . . . . . .|1

    Evaluate: . arctan(ln e) - arctan(ln 1)

    . . . . . .= . arctan(1) - arctan(0)

    . . . . . .= . π - 0 . = . π

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    It appears that the author didn't back-substitute.

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  4. #4
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    I made a mistake.
    I substituted for the derivative rather than original function.

    It seems the author made the same mistake I did.
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  5. #5
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    Quote Originally Posted by Soroban View Post
    Hello, ^_^Engineer_Adam^_^!

    I don't agree with the book either . . .


    . . . . . . . . . . . .1 . . . . dx
    We have: .∫ ----------------
    . . . . . . . . .1 + (ln x) . x

    . . . . . . . . . - . . . . . . dx
    Let u = ln x . . du = ---
    . . . . . . . . . . . - . - . . .x

    . . . . . . . . . . . du
    Substitute: .∫ -------- . = . arctan(u)
    . . . . . . . . . . 1 + u

    . . . . - . . . . - . . . - . . . . . . .|e
    Back-substitute: .arctan(ln x) |
    . . . . - . . . . - . . . - . . . . . . .|1

    Evaluate: . arctan(ln e) - arctan(ln 1)

    . . . . . .= . arctan(1) - arctan(0)

    . . . . . .= . π - 0 . = . π

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    It appears that the author didn't back-substitute.
    OK lets try integrating this numericaly. I won' go into the detail of
    how to do it, but the answer is ~=0.785, while the quoted answer is
    numericaly ~=0.433.

    RonL
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  6. #6
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    Really sorry

    the answer is really {\pi}/{4}...
    Made a Mistake again been careless with posting lately...
    sorry and ill try to post correctly next time..................
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  7. #7
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    Just in case
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