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Thread: Integration by substitution

  1. #1
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    Integration by substitution

    Hi

    I have the following integral but I cant see how to solve it, I know it involves substitution.

    $\displaystyle \int\dfrac{1}{\sqrt{x^2-4}}$

    Any hints would be appreciated

    James
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  2. #2
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    Try $\displaystyle x=2sec(\theta)$
    Do not forget the $\displaystyle dx$ in your integral.
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  3. #3
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    Hi

    I tried using it but didn't get very far, any chance you give me any further hints. The substitution I've done so far hasn't used trig but the standard $\displaystyle u$.

    Thanks
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  4. #4
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    When you substitute $\displaystyle x=2sec\theta$, you will face ( Assuming $\displaystyle tan\theta > 0$ ):
    $\displaystyle \int \frac{2 sec\theta tan\theta}{2 tan\theta} d\theta$

    You can not solve it?
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  5. #5
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    Substituting $\displaystyle u=\sqrt{x^2 - 4}$ will give another integral requires a trigonometric substitution.
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  6. #6
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    Hi

    In our handbook there is a list of standard integrals, this one gives

    $\displaystyle \int\dfrac{1}{\sqrt{x^2-a^2}}=\ln(x+\sqrt{x^2-a^2})$

    I could use this but they obviously want to see workings, any ideas how they got this?

    James
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  7. #7
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    By the same substitution $\displaystyle x=asec\theta$ .
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  8. #8
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    Thanks, got it, sorry it took a while.
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