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Thread: Integration of sqrt(1+t^2)dt

  1. #1
    Member Greengoblin's Avatar
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    Integration of sqrt(1+t^2)dt

    How can I integrate: $\displaystyle \int^1_{0}\sqrt{1+t^2}dt$?

    The obvious choice seems like substitution, but I got a bit stuck doing it that way:

    $\displaystyle u=1+t^2$
    $\displaystyle \frac{du}{dt}=2t$
    $\displaystyle du=2tdt$

    If it were $\displaystyle du=2dt$, then I could just multiply the limits by 2, but since there is a $\displaystyle t$ in there too, I'm not sure what to do, since it leaves the expression still with the unknown, $\displaystyle t$, in there.

    How can I do this?

    Thanks
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by Greengoblin View Post
    How can I integrate: $\displaystyle \int^1_{0}\sqrt{1+t^2}dt$?

    The obvious choice seems like substitution, but I got a bit stuck doing it that way:

    $\displaystyle u=1+t^2$
    $\displaystyle \frac{du}{dt}=2t$
    $\displaystyle du=2tdt$

    If it were $\displaystyle du=2dt$, then I could just multiply the limits by 2, but since there is a $\displaystyle t$ in there too, I'm not sure what to do, since it leaves the expression still with the unknown, $\displaystyle t$, in there.

    How can I do this?

    Thanks
    Thats exactly why its not a good idea to sub $\displaystyle u = 1+t^2$!

    See the problem is the square root sign. We need to get rid of it... that is why choose a substitution that will get rid of it...

    So sub $\displaystyle t = \tan u \Rightarrow dt = \sec^2 u \, du$,

    $\displaystyle \int^1_{0}\sqrt{1+t^2}dt =\int^{\frac{\pi}4}_{0}\sec^3 u\, du$

    Now try again

    Note that you can also try the hyperbolic substitution $\displaystyle t = \sinh u$.
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Greengoblin View Post
    How can I integrate: $\displaystyle \int^1_{0}\sqrt{1+t^2}dt$?

    The obvious choice seems like substitution, but I got a bit stuck doing it that way:

    $\displaystyle u=1+t^2$
    $\displaystyle \frac{du}{dt}=2t$
    $\displaystyle du=2tdt$

    If it were $\displaystyle du=2dt$, then I could just multiply the limits by 2, but since there is a $\displaystyle t$ in there too, I'm not sure what to do, since it leaves the expression still with the unknown, $\displaystyle t$, in there.
    That's not really a problem : we also know that $\displaystyle u=1+t^2$ hence $\displaystyle t^2=u-1\implies t=\pm\sqrt{u-1}$. As $\displaystyle t\in[0,1]$, we can safely choose $\displaystyle t=+\sqrt{u-1}$. Substituting this in $\displaystyle \mathrm{d}u=2t\mathrm{d}t$, we get $\displaystyle \mathrm{d}u=2\sqrt{u-1}\mathrm{d}t$ that is to say $\displaystyle \mathrm{d}t=\frac{\mathrm{d}u}{2\sqrt{u-1}}$. (Note that the easiest way to find this is to write $\displaystyle t=\sqrt{u-1}$ and to differentiate with respect to $\displaystyle u$.)

    Using this substitution, the original integral becomes $\displaystyle \int_1^2\frac{u}{2\sqrt{u-1}}\,\mathrm{d}u$... which is not easier to evaluate than the previous one.
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  4. #4
    Member Greengoblin's Avatar
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    Well if it is no easier, then I'll go for the first method mentioned.

    Can you please explain where the $\displaystyle tanu$ came form?
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    Lord of certain Rings
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    Quote Originally Posted by Greengoblin View Post
    Well if it is no easier, then I'll go for the first method mentioned.

    Can you please explain where the $\displaystyle tanu$ came form?
    As I mentioned previously, we have to get rid of the square root. That means we need to substitute such that 1+t^2 is some well known square function.

    I remember the identities $\displaystyle 1+\tan^2 u = \sec^2 u$ and $\displaystyle 1+ \sinh^2 u = \cosh^2 u$. These identities suggest that I put $\displaystyle t = \tan u$ or $\displaystyle \sinh u$
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Greengoblin View Post
    How can I integrate: $\displaystyle \int^1_{0}\sqrt{1+t^2}dt$?

    The obvious choice seems like substitution, but I got a bit stuck doing it that way:

    $\displaystyle u=1+t^2$
    $\displaystyle \frac{du}{dt}=2t$
    $\displaystyle du=2tdt$

    If it were $\displaystyle du=2dt$, then I could just multiply the limits by 2, but since there is a $\displaystyle t$ in there too, I'm not sure what to do, since it leaves the expression still with the unknown, $\displaystyle t$, in there.

    How can I do this?

    Thanks
    Have you considered a hyperbolic substitution?

    $\displaystyle t=\sinh(u)$

    Then:

    $\displaystyle \int^1_{0}\sqrt{1+t^2}~dt=\int_0^{\sinh^{-1}(1) }\cosh^2(u) ~du$


    RonL
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