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Math Help - Integration by substitution.

  1. #1
    Member Furyan's Avatar
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    Integration by substitution.

    Hello I'm sorry to ask again, but would somone be willing to look over this and tell me where I've gone wrong. Thank you.

    I know the method I'm using is probably very shoddy, but it's the best I could do.

    The question is integrate the following using the given substitution.

    \int\dfrac{\sqrt{x^2 + 4}}{x} dx

     u^2 = x^2 + 4

    I got \dfrac{dx}{du} = \dfrac{u}{x}

    I substituted these in first and got:

    \int(\dfrac{u}{x})(\dfrac{u}{x}) du

    \int\dfrac{u^2}{x^2}

    Then using  u^2 = x^2 + 4, I substituted again and got.

    \int\dfrac{u^2}{u^2 - 4}

    I didn't know how to integrate that so I used partial fractions and got:

    \int1 + \dfrac{1}{u - 2} - \dfrac{1}{u + 2}

    u + \ln(u - 2) - \ln(u + 2) + c

    Using u = \sqrt{x^2+4}, I got:

    \sqrt{x^2 + 4} + \ln\dfrac{\sqrt{x^2+4} -2}{\sqrt{x^2 + 4} + 2} + c

    The given answer is:

    \sqrt{x^2 + 4} - \ln\dfrac{\sqrt{x^2+4} + 2}{\sqrt{x^2 - 4} - 2} + c, which according to my graphing calculator is very different. I understand that this might be too much too ask, especially as I have only shown an outline of my working, but if anyone one can see an obvious place we're I've gone wrong I'd really appreciate it. My graphing calulator tells me the partial fractions are correct, so I really don't see what is wrong with what I have done.

    Thank you.
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  2. #2
    MHF Contributor

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    Re: Integration by substitution.

    Quote Originally Posted by Furyan View Post
    Hello I'm sorry to ask again, but would somone be willing to look over this and tell me where I've gone wrong. Thank you.

    I know the method I'm using is probably very shoddy, but it's the best I could do.

    The question is integrate the following using the given substitution.

    \int\dfrac{\sqrt{x^2 + 4}}{x} dx

     u^2 = x^2 + 4

    I got \dfrac{dx}{du} = \dfrac{u}{x}



    I substituted these in first and got:

    \int(\dfrac{u}{x})(\dfrac{u}{x}) du

    \int\dfrac{u^2}{x^2}

    Then using  u^2 = x^2 + 4, I substituted again and got.

    \int\dfrac{u^2}{u^2 - 4}

    I didn't know how to integrate that so I used partial fractions and got:

    \int1 + \dfrac{1}{u - 2} - \dfrac{1}{u + 2}

    u + \ln(u - 2) - \ln(u + 2) + c

    Using u = \sqrt{x^2+4}, I got:

    \sqrt{x^2 + 4} + \ln\dfrac{\sqrt{x^2+4} -2}{\sqrt{x^2 + 4} + 2} + c

    The given answer is:

    \sqrt{x^2 + 4} - \ln\dfrac{\sqrt{x^2+4} + 2}{\sqrt{x^2 - 4} - 2} + c, which according to my graphing calculator is very different. I understand that this might be too much too ask, especially as I have only shown an outline of my working, but if anyone one can see an obvious place we're I've gone wrong I'd really appreciate it. My graphing calulator tells me the partial fractions are correct, so I really don't see what is wrong with what I have done.

    Thank you.
    Look at this.
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  3. #3
    Member Furyan's Avatar
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    Re: Integration by substitution.

    Hello Plato,

    Thank you for your reply.

    Well apart from the fact that they all start with \sqrt{x^2 + 4}, they are all quite different. Well it's some consolation that Wolfram's answer is different to the given answer, but then I still don't see where I've gone wrong . Just out of interest why is Wolfram using \log_{10}.

    Apparently 'small animal care' is under-subscribed at my college, I think I might migrate.
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  4. #4
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    Re: Integration by substitution.

    Quote Originally Posted by Furyan View Post
    Well apart from the fact that they all start with \sqrt{x^2 + 4}, they are all quite different. Well it's some consolation that Wolfram's answer is different to the given answer, but then I still don't see where I've gone wrong . Just out of interest why is Wolfram using \log_{10}.

    They are not using \log_{10}.

    Like most modern practice in mathematics we use \log(x) in place of the out-of-date \ln(x).

    See a good calculus textbook like Gillman & McDowell.
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  5. #5
    Member Furyan's Avatar
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    Re: Integration by substitution.

    Right. Thank you for the book recommendation.

    I have just checked the three solutions again. The one I got is, in fact, equivalent to the one given by Wolfram

    Thank you for the link to that solution.
    Last edited by Furyan; February 12th 2013 at 02:01 AM.
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  6. #6
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    Re: Integration by substitution.

    More to the point, look here.

    I.e. your substitution is fine (though completely different from Wolfram's), just see 'alternate forms' no. 1. (Multiplying top and bottom by the top.)

    Probably the sign typo here, though...

    Quote Originally Posted by Furyan View Post
    The given answer is:

    \sqrt{x^2 + 4} - \ln\dfrac{\sqrt{x^2+4} + 2}{\sqrt{x^2 - 4} - 2} + c
    ... isn't helping.

    PS: oops, you just noticed anyway!
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  7. #7
    Member Furyan's Avatar
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    Re: Integration by substitution.

    Thank you for your efforts. It's helpful, and good to know of other ways in which I can check my answers. Most of the time the given answer is correct and I'm not, even if I can't usually see where I've gone wrong.

    Thanks again.
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