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Math Help - Indefinite Integration

  1. #1
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    Indefinite Integration

    The question:

    \int \frac{dx}{4 + x^2}

    I'm fairly sure I have to use some sort of substitution for this question. But I can't seem to get it to work. >_<
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  2. #2
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    No substitution required, use the fact that

    \int \frac{a}{a^2 + x^2}~dx = Tan^{-1}\left(\frac{x}{a}\right)+C
    Last edited by mr fantastic; October 27th 2010 at 10:16 PM. Reason: Inserted brackets.
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  3. #3
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    Quote Originally Posted by pickslides View Post
    No substitution required, use the fact that

    \int \frac{a}{a^2 + x^2}~dx = Tan^{-1}\left(\frac{x}{a}\right)+C
    What is this relationship called? I'm a little confused.
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  4. #4
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    This is just a formula for the indefinite integral

    here is where it comes from

    \displaystyle \int \frac{a}{x^2+a^2}dx

    let x=a\tan(\theta) \implies dx=a\sec^2{\theta}d\theta this gives

    \displaystyle \int \frac{a}{a^2\tan^2(\theta)+a^2}a\sec^2{\theta}d\th  eta=\int d\theta =\theta=\tan^{-1}\left(\frac{x}{a} \right)+c
    Last edited by TheEmptySet; October 28th 2010 at 09:20 AM.
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  5. #5
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    Ahh, ok. Is there other similar formulas I should be aware of?

    Also, I don't understand how the numerator is considered 'a' in my question when the denominator has '4', which is not a^2
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  6. #6
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    Or \displaystyle \int \frac{1}{x^2+a^2}\;{dx} = \frac{1}{a}\tan^{-1}{\frac{x}{a}}+k (same thing).

    Or letting x = 2\tan{x} in the OP's integral (same thing again).
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  7. #7
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    Quote Originally Posted by Glitch View Post
    Ahh, ok. Is there other similar formulas I should be aware of?
    There are quite a few.

    Here's some that are similar

    \int \frac{1}{\sqrt{a^2-x^2}}~dx = Sin^{-1}\left(\frac{x}{a}\right)+C

    \int \frac{-1}{\sqrt{a^2-x^2}}~dx = Cos^{-1}\left(\frac{x}{a}\right)+C
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  8. #8
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    Quote Originally Posted by Glitch View Post
    Also, I don't understand how the numerator is considered 'a' in my question when the denominator has '4', which is not a^2
    It doesn't matter since we can write any number k as (\sqrt{k})^2. For example, 4 = (\sqrt{4})^2 = 2^2.
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