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Thread: Intergration help

  1. #1
    Super Member 11rdc11's Avatar
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    Intergration help

    I'm having problems with these 2 integrals and any help would be great.

    $\displaystyle \int\frac{dx}{\sqrt{x^2-9}}$

    here is my work

    $\displaystyle x = 3\sec{\theta}$

    $\displaystyle dx = 3\tan{\theta}\sec{\theta}$

    plugging that in

    $\displaystyle \int \frac{3\tan{\theta}\sec{\theta}d\theta}{3\tan{\the ta}}$

    $\displaystyle \int \sec{\theta}d\theta$

    $\displaystyle \ln{|\sec{\theta}+\tan{\theta}|}$

    $\displaystyle \ln|\frac{x}{3} + \frac{\sqrt{x^2-9}}{3}| + C$

    but the book has

    $\displaystyle \ln|x+ \sqrt{x^2-9}| + C$

    where did I go wrong?


    The other problem is

    $\displaystyle \int \frac{dx}{(4x^2-9)^3}$

    I know I have to divide

    $\displaystyle 4x^2-9$

    by 4 to get

    $\displaystyle x^2 - \frac{9}{4}$

    but what do I divide the numerator by then, 64?

    Thanks in advance
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  2. #2
    Junior Member Evales's Avatar
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    Re: integration help

    Hi,
    I can't figure out what is going on with the first integral. I plugged it into Wolfram Alpha which gives an alternative answer which is quite confusing.

    However with your second one yes, simply divide the numerator/ whole integral by 64.
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  3. #3
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    I solved the first problem and got the same answer you did. I used Wolfram Mathematica to confirm it, and your answer is correct. I guess the book answer is wrong.

    Patrick
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  4. #4
    Super Member 11rdc11's Avatar
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    ok thanks but I'm still having problems on the last problem

    $\displaystyle \int \frac{dx}{(4x^2-9)^3}$

    $\displaystyle \int \frac{\frac{dx}{64}}{(x^2-\frac{9}{4})^3}$

    $\displaystyle x = \frac{2}{3}\sec{\theta}$

    $\displaystyle dx = \frac{2}{3}\sec{\theta}\tan{\theta}$

    plugging all that in I get

    $\displaystyle \frac{2}{2187} \int \frac{\sec{\theta}\tan{\theta}d\theta}{\tan^6{\the ta}}$

    and this is where I get stuck. Could someone check my work and show me how to solve it? Thanks in advance.
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  5. #5
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by PatrickFoster View Post
    I solved the first problem and got the same answer you did. I used Wolfram Mathematica to confirm it, and your answer is correct. I guess the book answer is wrong.

    Patrick
    I figured out what the book did.

    $\displaystyle \ln|\frac{x+\sqrt{x^2-9}}{3}|+ C$

    $\displaystyle \ln{|x+\sqrt{x^2-9}|}-\ln{3}+C$

    they just combine the constant of ln3 with the constant.
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