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 April 11th, 2009, 09:22 AM #1 Senior Member   Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0 Need help for an honest example of the integral of (exp(x)cos(x)) one of the way to solving this demonstrate the use of "back and forward" between the equal sign the integral of (exp(x)sin(x)cos(x) one of the way to solving this demonstrate the use of "up and down" between spaces the integral of (x*x^x*x^x^x*x^x*x) one of the way to solving this demonstrate the use of "a central" between expressions Give one example of a integral that demonstrate the use of "up and down," "back and forward" and then a central soft clap of hands (not sequentially in that order)?
 April 11th, 2009, 02:01 PM #2 Guest   Joined: Posts: n/a Thanks: Re: Need help for an honest example of Maybe some one else knows, but I do not know what you're talking about. Here is one way to find the integral of $e^{x}cos(x)$ using parts. $uv-\int vdu$ $\int e^{x}cos(x)dx$ Let $u=e^{x}, \;\ dv=cos(x)dx, \;\ du=e^{x}dx, \;\ v=sin(x)$ $e^{x}sin(x)-\int e^{x}sin(x)dx$ Do it again: $\int e^{x}sin(x)dx$ Let $u=e^{x}, \;\ dv=sin(x)dx, \;\ du=e^{x}dx, \;\ v=-cos(x)$ $-e^{x}cos(x)+\int e^{x}cos(x)dx$ Put it together with the first one: $e^{x}sin(x)-\left[-e^{x}cos(x)+\int e^{x}cos(x)dx\right]$ $\int e^{x}cos(x)dx=e^{x}sin(x)+e^{x}cos(x)-\int e^{x}cos(x)dx$ Add $\int e^{x}cos(x)dx$ to both sides: $2\int e^{x}cos(x)dx=e^{x}sin(x)+e^{x}cos(x)$ $\int e^{x}cos(x)dx=\frac{e^{x}sin(x)}{2}+\frac{e^{x}cos (x)}{2}$ $=\fbox{\frac{e^{x}(sin(x)+cos(x))}{2}}$
April 12th, 2009, 07:34 AM   #3
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Re: Need help for an honest example of

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 Originally Posted by galactus Maybe some one else knows, but I do not know what you're talking about.
It must have been a miscommunication. Because the second integral was explain to me by some guy from Copenhagen, sorry can't spell. Anyway, it simple once one realize what going on. There are complex representations of sin(x) and cos(x). Thus, one just move up or out to that space to take advantage of its algebra. Then move back via taking only the real part. However it is only formal symbolism but with good result that require analysis agrs along the way. It is fun to study these but at the end of the day, when one want to compute something one still have to expand it into series or limit of expression or the likes, it service are momental because it is the "algebra of calculus." For example, it is often state that exp(x^2) is not integralble into "elementary" functions. But indeed we have observe that it integral over the "entire" real is known; moreover,

exp(x^2) = cos(ix^2)-isin(ix^2) and so it too can symbolically have representation in term of "elementary" functions. The "movement" between such real and "imaginary" function should be study in details. It is like opening a fan (breaking things into pieces, like light spectral in physics), use it to cool one self, then close it (change the expression back to real or known function.) Thus, we have two different series representation of such integral (the other one being the interchange of limit after exp(x^2) are written in it series form.)

 April 13th, 2009, 07:09 AM #4 Senior Member   Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0 Re: Need help for an honest example of [quote="MyNameIsVu"]the integral of (x*x^x*x^x^x*x^x*x) one of the way to solving this demonstrate the use of "a central" between expressions quote] Other method of integration one should consider beside movement between R and C. One can also consider movement between R and R^b, like what mention above for the integration of e^(x^2) for b = 2. Another idea is: say e^x/x, then we should consider e^(x^2+y^2)/(xy). Still yet, another is: the role of the Laplace transform change differential equations into one that is algebraic, solve it there and then transfer it back. Thus, essentially on the symbolic level (as in mathematical condition to be fill in later) an integral can be equated to a differential equation or it matrix representation, then tranfers to the Algebraic realm via the mentioned transform to be solve then transfer accordingly. Still yet, the familar technique of "embeding an unity" can be employed. . . .
April 13th, 2009, 11:17 AM   #5
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Re: Need help for an honest example of

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 Originally Posted by MyNameIsVu the integral of (exp(x)cos(x)) one of the way to solving this demonstrate the use of "back and forward" between the equal sign
It occure to me while I was playing with my neighbor baby that in and every method, it is only a beginning of a chapter. For example, the above can be a vibration incrementally demised as the steps increased. Another thing to consider also is that the integral itself can have two equality as it form a two equations and two unknowns, and so on so forward using the "same" idea:

The integral of exp(x)(x^n)sin(x) [not exactly sure, don't wanna do the work ha ha ha],...

The integral of, well don't wanna think that much but yeah I guess it should be fun

And then the is there section on the vibration + the matrices (the variable are the integral.) ...

April 13th, 2009, 11:27 AM   #6
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Re: Need help for an honest example of

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 Originally Posted by MyNameIsVu the integral of (exp(x)sin(x)cos(x) one of the way to solving this demonstrate the use of "up and down" between spaces
Ok Ok, so we can also consider the vibration between spaces and such but that not consider another technique. Anyway, one can also consider the sending different part of the expression into more than one different spaces at the "same time." This is parameterizing as differ from parameterizing in the traditional ways, which in itself is another technique. So basically on can write a book just base on these 9 chapters [or 10 if you're really bold], more or less depending on the author arrangment, on the symbolic integration technique of one-variable (it is not useless.)

 April 13th, 2009, 01:06 PM #7 Senior Member   Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0 Re: Need help for an honest example of Well, the above is in actuality can be label as "book 1" of "N chapters symbolic integration techniques of one-variable." Book 2 focus on the "dx" or rather "dr." For example, one deduce the integration of power as it increase increment of "one" divided by that power with the usage of rectangular approximation. One can also consider peak as in just triangles at a point say "dt" taking the advantage of the underlying domain topological structures. This is moving between definition itself and require the student to take the so-called analysis version of the mentioned "book 1." But can be study symbolically by justly investigate it consistency with "dx" through computation. Look at icy snow flake it boundary could be that of a continuous "function" that is nowhere differentialbe (wrt "dx") yet in reality it should have an enclose area, and the nature of mathematics is to be as precise as possible. This is an example of the first chapter of "book 2." Edit: since the nature of manipulating definition itself, one can integrate relation and are no longer resticted to function only. The controling factor itself have a directive function within the differential, an idea is "df(r)" with a hollow circle enclosing the snow flake relation which in itself can be model by a limiting outter circle spiking like a winter snow. Note: the real snow is very different and it does not "look like a sun." Just an example to relate. Because each elongated peak are like that of fractal.
 April 13th, 2009, 05:42 PM #8 Senior Member   Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0 Re: Need help for an honest example of Two of the noticiable methods within integration techniques that require instant analysis is the so-called u-substitution and integration by part. The latter, one can refers to my other post on the product rule for one variable. With that being said, consider the integral of say x^3/(x^2-1), with the obvious substitution of, say, the denominator. The problem is with the singularities which result in the real-value function locally leaking into the complex plane, which is then noticiably C is not a field under such consideration. Thus one of the condition for the invoking the u-substitution "chapter" is the lack of singularities in "part 2 of book 1" kind of like a revisit but with even better precision. What are other condition/s, if any, for invoking such technique to maintain the "totalness" of the orginal expresssion? Other method to consider in the 1st part of book 1 is geometric techniques, and system contructions techniques. One example of the former is, say, finding arc length of curvature, one can just study the realisation of expression--as theirs "form" represent certain curve--and employ menthod such as "the computation of pi" post in the computer science section. The latter, one can just find an example of it in the construction of orthorgonal functions. Still yet, the 1st part is incomplete as linear operator have matrices approximations (as in one of William Averson idea, I think.) The way I understand it is, for example, of such approximation is the countability of the rational as they are "partition" into term within the "matrix," it seems like going in circle as in orthorgonal functions, but in actuality it is the "partition" of the domain and/or range itself.
 April 14th, 2009, 06:26 AM #9 Senior Member   Joined: Apr 2009 From: Mesa, Arizona Posts: 161 Thanks: 0 Re: Need help for an honest example of Part two of book 2 contains at least one* amazing technique of integration. Thus the informal symbolic 1st part is not in it entire grasp of "the symbolic integration techniques of one variable." I will not make mention of this technique here, but point out another technique which can be dubble as "leaf equality" or "knoted equality",.. and it has it distinction from that of the mentioned matrix integral-unknown expressions. For example, consider the integral of cos(sinx). It desciption is in general a long string of equality amongst the relation the span out horizontally and in some expression alone the way brances vertical peaks and downward ropes itself expand horizontally at some integral expression and so on so forward employing the first technique listed in the first post of this topic, each junctures is like that of the mentioned technique. Let start with the example, let s = sin(x) = cos(x-pi/4) = (sqrt(2)/2)cos(x) +(sqrt(2)/2)sin(x), thus cos(x) = (sqrt(2)-1)s. Hence we have the integral of a number that is multiply by the expression cos(s)/s. Now the application follow, say, through the use of integration by parts. Edit: I guess one should note that some technique is not really an technique for say since it have not been "generalised" (note, there is a different in abstractisation and generalisation) as in writting an entire chapter on it with later mention of the so-called "this is how far the author had travel" and label as "problem plus" and the likes. Here another, "technique" but have not been generalised: Say we have the string that have sin and cos involved at each end of the juncture, the ballance between the two can be meet at the middle via a substitution that mantain the likeness of whatever expression it is attached to. Edit 2: *Actually I believe the existence of at least two such techniques in the "2nd part of book 1."
April 14th, 2009, 09:09 AM   #10
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Re: Need help for an honest example of

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 Originally Posted by MyNameIsVu the integral of (exp(x)sin(x)cos(x) one of the way to solving this demonstrate the use of "up and down" between spaces
Moving between mathematical spaces are what consider elementary techniques within the context of topic such as this. For example is the first weilding or sewing method, as demonstrates for say, between the "realm of languages" and its manifestation in the "mathematical realm" in the following:

Find the mathematical expression for the following description of a function:

"The saw-like movement of peaks over the rationals with lower and upper peak differ by one."

This is the "application" of two things, first is the usage of the matrix-integral mentioned earlier with term as sin(x)/x and the second is the weilding by integral to "create" or find the mathematical expression as limitless as possible express by the above sentence. Think of it in term of spectrals (itself another integration technique.*) Thus the result is a region express by a function. Can one write a one-variable function that "move"?

*Already mentioned earlier in other post, it starting point is method such as P.F.D.,...

Edit: should had use the word "merky or silky region"

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