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Math Help - Integration by Substitution Question

  1. #1
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    Integration by Substitution Question

    I have calculated this problem multiple times, but never obtain the same answer that Maple provides.

    The integration limits are from 1 to 4, and the integrand is e raised to the square root of x, divided by the root of x.
    I have substituted "u" for the upper square root of x, which leaves me with the new integral with an integrand of e raised to the "u", divided by 2, with integration limits from 1 to 2.

    When I evaluate this, the answer I obtain is approximately 2.3354. Maple, however, provides me with an answer of approximately 9.3415.

    This question has me banging my head against the wall , because I feel that I am right, but technology is telling me I am wrong!

    If someone could tell me what I am doing wrong, or what Maple might be doing, that would be very much appreciated. I apologize for my lack of mathematical notation. If someone could please tell me how to do the symbols on here, that would be awesome!
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  2. #2
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    Make the following substitution



    to transform the integral into

    .

    Maple is correct.
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  3. #3
    Super Member TheChaz's Avatar
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    I'll just elaborate a bit on what was shown above.
    You didn't change the LIMITS of integration when making the substitution. Notice that "Prove It" manually included the "u =..." in the limits. If you wanted to avoid making this mistake ever again, you could explicitly write "x = " or "u = " for each limit of integration. But eventually, you'll get used to this kind of thing and will succeed without such helps.
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  4. #4
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    Where does the 2 out front come from? If the upper root of x is u, its derivative will be 1/2*root of x. And thus, 1/root of x will be replaced by 1/2du, and the 1/2 can be pulled outside of the integral, leaving 1/2 multiplied by the integral of e^u, evaluated from 1 to 2, right? Where am I wrong in my thinking?
    By the way, how are you all doing fancy notation in the forum?
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  5. #5
    Super Member TheChaz's Avatar
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    Quote Originally Posted by SC313 View Post
    ...
    By the way, how are you all doing fancy notation in the forum?
    Online LaTeX Equation Editor - create, integrate and download is how I do it.


    The 2 out in front (in the first line) isn't the way most people would work this; it would appear later. If you're having trouble with getting the integrand to contain du (exactly), you might need to review u-substitution.
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    If u = Sqrt[x], then

    du/dx = 1/(2Sqrt[x])

    du = 1/(2Sqrt[x])dx

    2du = 1/(Sqrt[x])dx.


    So you don't replace 1/(Sqrt[x])dx with (1/2)du, you replace it with 2du.
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    I had everything right except it would be replaced by 2*du. I feel stupid! Sorry all!
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  8. #8
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    Quote Originally Posted by TheChaz View Post
    Online LaTeX Equation Editor - create, integrate and download is how I do it.


    The 2 out in front (in the first line) isn't the way most people would work this; it would appear later. If you're having trouble with getting the integrand to contain du (exactly), you might need to review u-substitution.
    Don't speak for most people...
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  9. #9
    Super Member TheChaz's Avatar
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    Quote Originally Posted by Prove It View Post
    Don't speak for most people...
    The thing is... with a statement like "most people..." it can be quite burdensome to disProve It.

    Maybe there's a way you could get the point across with using the imperative...
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  10. #10
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    Thanks all! Do either of you know of a really good book which deals with applications of calculus, such as mixture problems?
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  11. #11
    Super Member TheChaz's Avatar
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    Quote Originally Posted by SC313 View Post
    Thanks all! Do either of you know of a really good book which deals with applications of calculus, such as mixture problems?
    I try to steer clear of any sort of application of math! Wish I could be more help...
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  12. #12
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    Quote Originally Posted by TheChaz View Post
    I try to steer clear of any sort of application of math! Wish I could be more help...
    I hate it as well, but the teachers I have been graced with all adore applications. Well, what's a really good calculus book in general?
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  13. #13
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    Quote Originally Posted by SC313 View Post
    I hate it as well, but the teachers I have been graced with all adore applications. Well, what's a really good calculus book in general?
    Many forum users (here and elsewhere) like Amazon.com: Calculus, 4th edition (9780914098911): Michael Spivak: Books
    (Spivak's Calculus). I personally like Amazon.com: A Tour of the Calculus (9780679747888): David Berlinski: Books , but I've heard more criticism about this book than praise!
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  14. #14
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    Integration by Substitution Question

    Thanks!
    Last edited by mr fantastic; April 19th 2011 at 05:13 AM. Reason: Mved new questions to new thread.
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