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Math Help - How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

  1. #1
    Gui
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    How do I take this definite integral? From x^2 to sin(x), of t / t^4 + 1, dt.

    Hey, I'd appreciate any help taking the integral below. The two methods I know are by substitution and by parts, can I use either of those? Thanks in advance!

    \int_{x^2}^{sin(x)} \frac{t}{t^4 + 1}dt
    Last edited by Gui; June 14th 2012 at 03:36 PM.
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    The substitution u=t^2 works.
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    Quote Originally Posted by a tutor View Post
    The substitution u=t^2 works.
    yes it works and you get \arctan{t^2}|^{\sin x}_{x^2}. But, usually question like this ask to find the derivative of this term, do not it?
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    Gui
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    a tutor: I'm trying, but stuck. I'll try and get a scan of my results up.

    Kmath: Isn't it arctan(t^2)/2? I put it into my ti-89 and it gave me that; wolframalpha times out.
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    Quote Originally Posted by Gui View Post
    a tutor: I'm trying, but stuck. I'll try and get a scan of my results up.

    Kmath: Isn't it arctan(t^2)/2? I put it into my ti-89 and it gave me that; wolframalpha times out.
    yes of course, but I neglected 1/2.
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    Gui
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    Ah ok Here's my work so far, but I'm pretty much stuck here... I don't know what to do with the logs. Thanks in advance for all your help!

    How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.-image-19-.jpg

    How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.-image-20-.jpg
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    How did you deduce the last move of the first page?
    Are you sure that the question need not the derivative?
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    Gui
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    Kmath: Yeah, the question does ask for the derivative. Shouldn't I actually find f(x) though, before finding f(x)?

    The last move of the first page.... well, the derivative of ln(x) is 1/x, so the antiderivative of 1/(u^2 + 1) is ln(u^2 + 1)/derivative of u^2 +1, which is ln(u^2 + 1)/2u. Is that not correct?
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    [QUOTE=Gui;723170]Kmath: Yeah, the question does ask for the derivative. Shouldn't I actually find f(x) though, before finding f(x)?

    So you can find the derivative with out finding f(x). In this case you can use the fundamental theorem of calculus.
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    The last move of the first page.... well, the derivative of ln(x) is 1/x, so the antiderivative of 1/(u^2 + 1) is ln(u^2 + 1)/derivative of u^2 +1, which is ln(u^2 + 1)/2u. Is that not correct?[/QUOTE]

    May be you forgot to use the "Quotient rule" that is (\frac{f}{g})^\prime=\frac{f^\prime g-fg^\prime}{g^2}.
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    Gui
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    [QUOTE=Kmath;723173]
    Quote Originally Posted by Gui View Post
    Kmath: Yeah, the question does ask for the derivative. Shouldn't I actually find f(x) though, before finding f(x)?

    So you can find the derivative with out finding f(x). In this case you can use the fundamental theorem of calculus.
    Sorry, I didn't go over this in class. From reading the wiki page, I'm thinking it's the First Theorem I should be using? How would I apply it to the problem?
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    Re: How do I take this definite integral? FRom x^2 to sin(x), of t / t^4 + 1, dt.

    [QUOTE=Gui;723177]
    Quote Originally Posted by Kmath View Post

    Sorry, I didn't go over this in class. From reading the wiki page, I'm thinking it's the First Theorem I should be using? How would I apply it to the problem?
    Ok, I did not find what I meant in Wikipedia. Any way, here is roughly the Theorem

    \frac{d}{dx}\int_{f(x)}^{g(x)}h(t)dt=h(f(x))f^{\pr  ime}(x)-h(g(x))g^{\prime}(x).
    So, in your case we have

    \frac{\sin{x}}{1+{\sin^4 x}}\cos x-\frac{x^2}{1+x^8}(2x).


    You may find the theorem in any calculus book.
    Last edited by Kmath; June 14th 2012 at 04:47 PM.
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