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Math Help - Introduction to Calculus Tutorial

  1. #31
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    Quote Originally Posted by ThePerfectHacker View Post

    Find the integrals.

    4*) \int e^{\sqrt{x}} It has a star what do you think!?


    10)  \int \frac{1}{1-e^{2x}}dx Hard
    hi
    i cant seem to solv this two problems can u please provide solution to these?
    arrgh
    Last edited by mash; March 5th 2012 at 07:43 PM. Reason: fixed latex
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  2. #32
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    hi
    i cant seem to solv this two problems can u please provide solution to these?
    arrgh
    \int e^{\sqrt{x}} dx
    I will make the substitution u=\sqrt{x}. But if I do that, I always need to know what its derivative is, u' = \frac{1}{2\sqrt{x}}.

    If you look into the integral this factor does not appear. So what do we do? We make it appear. Multiply the numerator and denominator by this expression:

    \int e^{\sqrt{x}}\cdot 2\sqrt{x} \cdot \frac{1}{2\sqrt{x}} dx
    Now if we use u=\sqrt{x} then, u'=\frac{1}{\sqrt{x}} which is good because it appears as a factor and 2\sqrt{x} = 2u.
    Make the substitution,
    \int e^u \cdot 2u \cdot u' dx = \int 2u e^u du.
    You can now do this integral by parts.
    Last edited by mash; March 5th 2012 at 07:44 PM. Reason: fixed latex
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  3. #33
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    solution for number 10

    i get it now
    \int \frac{1}{1-e^{2x}}dx

    \int \frac{1 - e^{2x}+e^{2x}}{1-e^{2x}}dx

    \int \frac{1-e^{2x}}{1-e^{2x}}dx +  \frac{e^{2x}}{1-e^{2x}}dx

    x + \int \frac{e^{2x}}{u} \frac{du}{-2e^{2x}}

    x - \frac{1}{2}\ln(1-e^{2x}) +C

    but the integrator says

    x - \frac{1}{2}\ln(e^{2x}-1)
    hmm
    Last edited by mash; March 5th 2012 at 07:46 PM.
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  4. #34
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    i get it now
    \int \frac{1}{1-e^{2x}}dx

    \int \frac{1 - e^{2x}+e^{2x}}{1-e^{2x}}dx

    \int \frac{1-e^{2x}}{1-e^{2x}}dx +  \frac{e^{2x}}{1-e^{2x}}dx

    x + \int \frac{e^{2x}}{u} \frac{du}{-2e^{2x}}

    x - \frac{1}{2}\ln(1-e^{2x}) +C

    but the integrator says

    x - \frac{1}{2}\ln(e^{2x}-1)
    hmm
    Because you need to use | \, |.
    In that case,
     \ln |1-e^{2x}| = \ln |e^{2x}-1|
    Because,
    |a-b|=|b-a|
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  5. #35
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    just wanted to point out: isn't a sequence a function defined as  f: \mathbb{Z^{+}} \rightarrow A which means a sequence in the set  A ? So the codomain doesn't have to be  \mathbb{R} but can be an arbitrary set  A ? Actually  \mathbb{N} and  \mathbb{Z^{+}} are equivalent right? And a sequence is null when  \lim_{n \rightarrow \infty} f(n) = 0 when  \forall \epsilon \in \mathbb{R^{+}}, \exists N \in \mathbb{Z^{+}}, \forall n \in \mathbb{Z^{+}} (n \geq N \Rightarrow |f(n)| < \epsilon) . Also, I like to think of functions as follows: pretend you have a box subdivided into smaller boxes. These smaller boxes represent the elements of the codomain, and the points in the boxes represent the elements of the domain. So a function orders the points in the smaller boxes.
    Last edited by tukeywilliams; June 29th 2007 at 11:46 PM.
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  6. #36
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    Quote Originally Posted by tukeywilliams View Post
    just wanted to point out: isn't a sequence a function defined as  f: \mathbb{Z^{+}} \rightarrow A which means a sequence in the set  A ? So the codomain doesn't have to be  \mathbb{R} but can be an arbitrary set  A ? Actually  \mathbb{N} and  \mathbb{Z^{+}} are equivalent right? And a sequence is null when  \lim_{n \rightarrow \infty} f(n) = 0 when  \forall \epsilon \in \mathbb{R^{+}}, \exists N \in \mathbb{Z^{+}}, \forall n \in \mathbb{Z^{+}} (n \geq N \Rightarrow |f(n)| < \epsilon) . Also, I like to think of functions as follows: pretend you have a box subdivided into smaller boxes. These smaller boxes represent the elements of the codomain, and the points in the boxes represent the elements of the domain. So a function orders the points in the smaller boxes.
    I was talking about a "real sequence" in that case f:\mathbb{N}\to \mathbb{R}.
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  7. #37
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    TPH, your notes are extremely thorough and detailed, and are an excellent source for anyone studying Calculus. I hope you don't mind that I added some of my "simpler" Calculus notes to this forum. Like yours, my notes are not meant for students taking Advanced Calculus courses such as Real Analysis. I hope I am not stepping on your toes!
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  8. #38
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    Quote Originally Posted by colby2152 View Post
    TPH, your notes are extremely thorough and detailed, and are an excellent source for anyone studying Calculus. I hope you don't mind that I added some of my "simpler" Calculus notes to this forum. Like yours, my notes are not meant for students taking Advanced Calculus courses such as Real Analysis. I hope I am not stepping on your toes!
    that would be all good and well i believe. in fact, we encourage users to do things like that. that's why we have a mathwiki page (but it's not fully functional yet). as long as you're not "re-inventing the wheel" (that is, going over exactly the same stuff TPH covered, unless of course, you feel he left something out) i don't see why you can't post notes. you may want to do it in a different thread though, this is TPH's thread.
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  9. #39
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    Good Reference Textbook

    I believe the textbook

    Introduction to Stochastic Calculus with Applications, 1st Edition

    will be useful for you to study Mathematics

    Good Luck and wish help for you.

    hehe ^_^
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  10. #40
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    Just some spelling corrections: Differenciable, Differencial, Differenciation should all be Differentiable, Differential, Differentiation respectively. Basically, the C's should be T's.
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  11. #41
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    Quote Originally Posted by scorpion007 View Post
    Just some spelling corrections: Differenciable, Differencial, Differenciation should all be Differentiable, Differential, Differentiation respectively. Basically, the C's should be T's.
    Oh, that's just TPH, you shouldn't bother that much for it

    Another example that comes in mind is that he once said "diagnol" for "diagonal". It's common, but it doesn't alter the quality
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  12. #42
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    On the off chance that TPH or some moderator is still attending responses to this thread--

    Regarding elliptical integrals TPH dixit: "Now the problem is that this integral cannot be expressed in elementary functions (the standard functions) like I mentioned in the previous lecture that that sometimes happens. This problem was studied primarily by Abel and lead to something call Elliptic Functions. These are new types of functions that integrate this. I myself do not know what courses actually teach these functions, not because they are difficult but because it is unnecessary."

    And elsewhere, regarding the integration of radicals (particularly the arclength formula):

    "We did not discuss how to integrate that, it can be done. The types of functions used are hyperbolic functions, they are related to the exponental e^x function. But that is too advacned for us, and all we did was set up the integral."

    I would like to look into these related matters a little more deeply. Can TPH or anyone recommend a text please?

    No websites please, I've already been there.

    Oh yes, and many thanks indeed to the author of this awesome thread.
    Last edited by rainer; December 10th 2009 at 11:22 AM. Reason: added a second quote
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  13. #43
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    Quote Originally Posted by ThePerfectHacker View Post
    11*)Show that if \sum_{n=0}\frac{kn}{n^2+1} converges only when k=0.
    1-if k < 0 then the series diverges by the L.C.T with -1/n
    2-if k > 0 then the series diverges by the limit comparison test with 1/n

    If k=0 the series converges since  \sum_{}0 converges.

    but 1 is wrong
    since -1/n is not positive

    Help please!
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  14. #44
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by TWiX View Post
    1-if k < 0 then the series diverges by the L.C.T with -1/n
    2-if k > 0 then the series diverges by the limit comparison test with 1/n

    If k=0 the series converges since  \sum_{}0 converges.

    but 1 is wrong
    since -1/n is not positive

    Help please!
    you could write k = -m where m>0 and consider the series \sum_{n=0}^\infty \frac {mn}{n^2 + 1}. By 2 it diverges. Hence the series \sum_{n=0}^\infty \frac {kn}{n^2 + 1} = - \sum_{n=0}^\infty \frac {mn}{n^2 + 1} diverges.
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  15. #45
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    Quote Originally Posted by ThePerfectHacker View Post
    \int e^{\sqrt{x}} dx
    I will make the substitution u=\sqrt{x}. But if I do that, I always need to know what its derivative is, u' = \frac{1}{2\sqrt{x}}.

    If you look into the integral this factor does not appear. So what do we do? We make it appear. Multiply the numerator and denominator by this expression:

    \int e^{\sqrt{x}}\cdot 2\sqrt{x} \cdot \frac{1}{2\sqrt{x}} dx
    Now if we use u=\sqrt{x} then, u'=\frac{1}{\sqrt{x}} which is good because it appears as a factor and 2\sqrt{x} = 2u.
    Make the substitution,
    \int e^u \cdot 2u \cdot u' dx = \int 2u e^u du.
    You can now do this integral by parts.
    Sorry, But this is wrong
    You should prove that x dont equal to zero first
    you cant devide by a variable If you dont prove it dont equal to 0
    Substitute u=\sqrt{x}
    then
    u^2 = x
    dx=2udu
    \int e^{\sqrt{x}} dx = 2 \int u e^{u} du
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