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Thread: Definate integral calculation difficulty

  1. #1
    Member Furyan's Avatar
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    Definate integral calculation difficulty

    Hello,

    The question is to evaluate the following integral.

    $\displaystyle \int_{\tfrac{\pi}{4}}^{\tfrac{\pi}{2}}\ 7 \cot^2(7x) dx$

    by writing it as

    $\displaystyle 7\int_{\tfrac{\pi}{4}}^{\tfrac{\pi}{2}}\ \csc^2(7x) - 1 dx$

    I got

    $\displaystyle -\cot(7x) -7x$

    which I think is right, my difficulty seems to be with evaluating. The answer is given as $\displaystyle 1+\dfrac{7\pi}{4}$, but I keep getting $\displaystyle -1 - \dfrac{7\pi}{4}$.

    I'd be very grateful if someone would check this.

    Thank you.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Definate integral calculation difficulty

    Hi Furyan!

    You have a couple of singularities in there.
    Perhaps you could split the integral in such a way you can take care of these singularities?
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  3. #3
    Member Furyan's Avatar
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    Re: Definate integral calculation difficulty

    Quote Originally Posted by ILikeSerena View Post
    Hi Furyan!

    You have a couple of singularities in there.
    Perhaps you could split the integral in such a way you can take care of these singularities?
    Hi ILikeSerena, I certainly feel very close to a black hole, but I'm afraid I can't resist the field strength. I figure since I've got the integral it should just be plain sailing. What do you mean by singularities?

    Thank you
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Definate integral calculation difficulty

    Quote Originally Posted by Furyan View Post
    Hi ILikeSerena, I certainly feel very close to a black hole, but I'm afraid I can't resist the field strength. I figure since I've got the integral it should just be plain sailing. What do you mean by singularities?

    Thank you
    Actually, now that I'm taking a better look, it's even worse. Your integral does not exist.
    Your given answer is wrong.

    The problem is that $\displaystyle \cot 7x$ is not defined for all the points in the range.
    It has so called singularities in $\displaystyle x=0, \frac \pi 7, \frac {2\pi} 7, \frac {3\pi} 7$.

    It means that you have to split your integral up at these singularities into so called improper integrals (see included wiki link).
    Your integral expression $\displaystyle 7 \cot^2(7x)$ has infinitely high mountains in these singularities.
    Now it could still be possible that these mountains have a finite area, but in this case they don't.


    If we split up your integral at the singularities, we get:

    $\displaystyle \int_{\pi / 4}^{\pi / 2} 7 \cot^2(7x) dx = \int_{\pi / 4}^{2\pi / 7} 7 \cot^2(7x) dx + \int_{2\pi / 7}^{3\pi / 7} 7 \cot^2(7x) dx + \int_{3\pi / 7}^{\pi / 2} 7 \cot^2(7x) dx$


    Let's pick the first term:

    $\displaystyle \int_{\pi / 4}^{2\pi / 7} 7 \cot^2(7x) dx$
    $\displaystyle = \lim_{b~ \uparrow~ {2\pi / 7}} \int_{\pi / 4}^{b} 7 \cot^2(7x) dx $

    $\displaystyle = \lim_{b~ \uparrow~ {2\pi / 7}} (-\cot(7x)-7x)|_{\pi / 4}^b$

    $\displaystyle =\text{does not exist}$


    This improper integral does not exist, because $\displaystyle \cot(7\cdot {2\pi \over 7}) = \cot(2 \pi)$ is not defined.

    None of these improper integrals exist, so your integral does not exist either $\displaystyle \Box$.
    Last edited by ILikeSerena; Jan 26th 2013 at 01:29 AM.
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  5. #5
    Member Furyan's Avatar
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    Re: Definate integral calculation difficulty

    Quote Originally Posted by ILikeSerena View Post
    Actually, now that I'm taking a better look, it's even worse. Your integral does not exist.
    Your given answer is wrong.
    Hi IlikeSerena,

    Thank you for taking the time to post such a comprehensive reply. I now understand that the integral does not exist because you cannot integrate across a discontinuity. I also now see that it's important to sketch a graph when finding a definite integral and doing so helped me to see why I was getting a couple of other questions wrong. One thing that was confusing me was that when evaluating:

    $\displaystyle \dfrac{1}{\tan\tfrac{7\pi}{2}}$ I got an error

    but when evaluating

    $\displaystyle \dfrac{\cos\tfrac{7\pi}{2}}{\sin\tfrac{7\pi}{2}}$ I got an answer of 0

    I now have to evaluate:

    $\displaystyle \int_0^2\dfrac{5 - 2x}{(x - 1)(2x + 1)}$

    Integrating I get $\displaystyle \ln(x - 1) - 2\ln(2x + 1)$

    The given answer is $\displaystyle \ln\dfrac{1}{25}$

    Am I right in thinking this integral does not exist either since $\displaystyle y = \dfrac{1}{x - 1}$ is not defined when $\displaystyle x = 1?$

    Thank you
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Definate integral calculation difficulty

    Quote Originally Posted by Furyan View Post
    Hi IlikeSerena,

    Thank you for taking the time to post such a comprehensive reply. I now understand that the integral does not exist because you cannot integrate across a discontinuity. I also now see that it's important to sketch a graph when finding a definite integral and doing so helped me to see why I was getting a couple of other questions wrong.
    Good!


    One thing that was confusing me was that when evaluating:

    $\displaystyle \dfrac{1}{\tan\tfrac{7\pi}{2}}$ I got an error

    but when evaluating

    $\displaystyle \dfrac{\cos\tfrac{7\pi}{2}}{\sin\tfrac{7\pi}{2}}$ I got an answer of 0
    The point is that you cannot divide by zero (or something bad happens to the universe, like black holes and singularities and stuff ).

    The tan function is actually a division of the sine by the cosine.
    It does not exist if the cosine is zero, which is the case at 7pi/2, or more generally at pi/2 + k pi, where k is an integer.

    When the tan does not exist, you can't take its inverse any more, even if that would have existed as in your case.

    So $\displaystyle 1 \over \displaystyle ({\sin 7\pi/2 \over \cos 7\pi/2})$ does not exist, because it contains a division by zero.
    But $\displaystyle {\cos 7\pi/2 \over \sin 7\pi/2}$ does exist, since there is no division by zero.




    I now have to evaluate:

    $\displaystyle \int_0^2\dfrac{5 - 2x}{(x - 1)(2x + 1)}$

    Integrating I get $\displaystyle \ln(x - 1) - 2\ln(2x + 1)$

    The given answer is $\displaystyle \ln\dfrac{1}{25}$

    Am I right in thinking this integral does not exist either since $\displaystyle y = \dfrac{1}{x - 1}$ is not defined when $\displaystyle x = 1?$

    Thank you
    Yes, you are correct. The integral does not converge due to the singularity at x=1.


    Btw, your anti-derivative is not entirely correct, since it should also be defined for x < 1.
    You lost that somewhere in your integration.
    The anti-derivative you have only works for the part to the right of the singularity.
    You would probably need to find another anti-derivative for the left part.
    Last edited by ILikeSerena; Jan 26th 2013 at 11:34 AM.
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  7. #7
    Member Furyan's Avatar
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    Re: Definate integral calculation difficulty



    Yes, you are correct. The integral does not converge due to the singularity at x=1.


    Btw, your anti-derivative is not entirely correct, since it should also be defined for x < 1.
    You lost that somewhere in your integration.
    The anti-derivative you have only works for the part to the right of the singularity.
    You would probably need to find another anti-derivative for the left part.
    Thank you very much. Yes I can see that the anti-derivative only works for the part to the right of the singularity. For now I have no idea how I would go about finding another one for the left part, but I will think about it.

    Thanks again for all your help.
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