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Math Help - Quiz question that stumped me

  1. #1
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    Quiz question that stumped me

    We had a quiz today and there was a problem where I didn't even know where to start... I was hoping someone could help me out on it.

    I'm not very good with the syntax to this editor, but I'll try my best.

    Evaluate the following integral:

    ∫ x^4 / (x^10 + 16) dx

    Clarification:
    Numerator: x^4
    Denominator: x^10 + 16

    (My problem was I didn't know if I should try substitution, because it doesn't look like I can factor the denominator.)


    Thanks,
    -Freyr.
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  2. #2
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    Hello, Freyr!

    \int\frac{x^4}{x^{10} + 16}\,dx

    We have: . \int \frac{x^4\,dx} {(x^5)^2 + 16}

    Let u = x^5\quad\Rightarrow\quad du = 5x^4\,dx \quad\Rightarrow\quad x^4\,dx = \frac{du}{5}

    Substitute: . \int\frac{\frac{du}{5}}{u^2 + 16} \;=\;\frac{1}{5}\int\frac{du}{u^2+16} \quad\hdots Got it?

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  3. #3
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    Oh... Wow. That's really thinking outside the box. But when you do it, it looks so simple.

    So the integral would be 1/5 [ ln | u^2 + 16| ] ?
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  4. #4
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    Integrals

    NEVER use \ln when the denominator contains a quadratic expression. Use \ln only if it's LINEAR. You can always check your answer by differentiation.

    Recall that an integral of the type \int\frac{dx}{a^2+x^2} can be simplified by letting  x = a\ tan\ (\theta) . So in the end, the integral becomes  = \frac{1}{a} \arctan \frac{x}{a} + C.

    And note that, in this case,  a = 4, x = u . Either way, you're substituting twice, so do not use the variable x. Make sure your final answer contains x and not u or any other variable.

    I hope that helps.

    P.S. Maths ALWAYS requires you to think 'out of the box' . That's why it's so much fun.
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  5. #5
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    Quote Originally Posted by ILoveMaths07 View Post
    NEVER use \ln when the denominator contains a quadratic expression. Use \ln only if it's LINEAR. You can always check your answer by differentiation.

    [snip]
    So what would you suggest the answer to \int \frac{2x}{x^2 + 1} \, dx is equal to!?



    I understand what you're trying to say. But such sweeping generalisations are rarely true and serve only to create confusion.
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  6. #6
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    Red face

    Heh, that would be ln |x^{2}+1| + C. Umm... Then how would you generalise what I was trying to say? How would you put it in other words?

    ILoveMaths07.
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  7. #7
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    Quote Originally Posted by ILoveMaths07 View Post
    Heh, that would be ln |x^{2}+1| + C. Umm... Then how would you generalise what I was trying to say? How would you put it in other words?

    ILoveMaths07.
    You can't, unless you want a cookbook that rivals the American constitution for complexity.

    There are general principles but ultimately the student's ability and experience is what solves the problem. Especially experience.

    It also helps if the student has met and can remember the basic standard forms (like the one that appears in this thread). Unfortunately that's rarely the case.

    The mistake the OP made is a consequence of misunderstanding what is in the Cook Book Calculus For Dummies.
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