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Math Help - Partial Fraction

  1. #1
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    Partial Fraction

    Q:
    Attached Thumbnails Attached Thumbnails Partial Fraction-17.gif  
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  2. #2
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    Factor out a 9.

    Meaning,

    1/(9+x^2) can be written as

    1/9 * 1/(1+(x/3)^2)

    Now let t=x/3
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by qbkr21 View Post
    Q:
    Here
    Attached Thumbnails Attached Thumbnails Partial Fraction-arctan.gif  
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    Re:

    Thanks Guys!
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  5. #5
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    Hello, everyone!

    A question:

    Doesn't anyone teach the general formulas for inverse trig integrals?


    . . . . . .du
    . . ∫ ------------ . = . arcsin(u/a) + C
    . . . .√aČ - uČ


    . . . . . .du
    . . ∫ ---------- . = . (1/a)·arctan(u/a) + C
    . . . .uČ + aČ


    . . . . . . .du
    . . I ------------- . = . (1/a)·arcsec(u/a) + C
    . . . .u√uČ - aČ


    A half-century ago, my professor derived these formulas for us.
    . . We have never had to factor-out-4 or factor-out-9.


    [rant]

    I've made this query at other math sites.

    No one has ever thanked me for these streamlined formulas.

    Instead, someone replies with something like:
    "I'd rather have my students understand the concepts
    . . rather than memorizing a list of formulas"
    as if memorizing is a Bad Thing.

    My first response has been:
    If you've already memorized a formula with uČ + 1
    . . why not expand it to include the "a"?

    They usually reply with hints that I'm a formula-happy moron.
    (Have they ever seen any of my posts?
    Isn't Understanding the main feature of all my explanations?)

    And then I must respond with something like:
    If Understanding is the most important issue,
    . . then we must not memorize the Quadratic Formula.
    Instead, we make the student complete-the-square every time.

    Then someone says, "What's wrong with completing-the-square?
    I'd rather have them understand the procedure, than just memorize a formula."

    Then I must explain the Philosophy of Learning.
    When we learn a new fact, we in fact memorize it.
    . . Yes, we "commit it to memory" for future use.
    Then (and here's the important part) . . . we move on.
    We add the formula/shortcut to our arsenal (in the war against our own ignorance)
    . . and progress to bigger and better things.
    And it is hoped that we retain the reasoning behind the formulas/shortcuts.

    We do not draw a line in the sand and say, "This is far enough!
    . . From now on, everything will be derived ... no more blind memorization!"


    If Memorization is a misdemeanor, then 2 Ś 3 .= .6 is at least a felony.
    . . After all, it can derived from: 3 + 3 .= .6

    In fact, throw out the addition tables!
    . . {◊ ◊ ◊} + {◊ ◊ ◊} .= .{◊ ◊ ◊ ◊ ◊ ◊}
    . . . 1-2-3 . . . . 1-2-3 . - . . .1*2*3-4-5*6

    We can always stop and "count apples", right?

    [/rant]

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  6. #6
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    You don't need partial fraction

    You don't need partial fraction. Just use the substitution rule.
    Last edited by CaptainBlack; May 17th 2007 at 11:38 PM.
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  7. #7
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    Re:

    Re:
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  8. #8
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    The integral can be found as follows

    The integral can be evaluated as follows:
    Last edited by CaptainBlack; May 17th 2007 at 11:38 PM.
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  9. #9
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    Quote Originally Posted by Soroban View Post
    Hello, everyone!

    A question:

    Doesn't anyone teach the general formulas for inverse trig integrals?


    . . . . . .du
    . . ∫ ------------ . = . arcsin(u/a) + C
    . . . .√aČ - uČ


    . . . . . .du
    . . ∫ ---------- . = . (1/a)·arctan(u/a) + C
    . . . .uČ + aČ


    . . . . . . .du
    . . I ------------- . = . (1/a)·arcsec(u/a) + C
    . . . .u√uČ - aČ
    That's exactly I was to write.

    Thanks Soroban.
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  10. #10
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    partial fraction

    partial fraction
    Attached Thumbnails Attached Thumbnails Partial Fraction-17may2007.jpg  
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