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| In [[calculus]], the [[indefinite integral]] of a given function (i.e., the set of all [[antiderivative]]s of the function) is only defined [[up to]] an additive constant, the '''constant of integration'''.<ref>{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals |publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=0-495-01166-5}}</ref><ref>{{cite book | last1=Larson | first1=Ron | authorlink=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=[[Brooks/Cole]] | edition=9th | year=2009 | isbn=0-547-16702-4}}</ref> This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function <math>f(x)</math> is defined on an [[interval (mathematics)|interval]] and <math>F(x)</math> is an antiderivative of <math>f(x)</math>, then the set of ''all'' antiderivatives of <math>f(x)</math> is given by the functions <math>F(x) + C</math>, where ''C'' is an arbitrary constant. The constant of integration is sometimes omitted in [[lists of integrals]] for simplicity.
| | If your computer is running slow, you have probably gone from the different stages of rage plus frustration. Having such a excellent tool like a computer could seem like a curse along with a blessing simultaneously whenever this happens. It is fantastic when it is actually running rapidly and smooth, nevertheless then when it begins acting strange and slows way down, frustration sets inside. How will something because wonderful because a computer make a person so mad?<br><br>We all understand which the registry is the important component of the Windows running program as it stores all info about the Dll files, programs on the computer plus system settings. However, as days by, it really is unavoidable which we could encounter registry matter due to a huge amount of invalid, useless plus unwanted entries.<br><br>H/w connected error handling - when hardware causes BSOD installing newest fixes for the hardware and/ or motherboard will help. We could additionally add fresh hardware which is compatible with all the system.<br><br>There can be several factors why a computer can lose speed. Normal computer utilize, such as surfing the Internet may receive a running system in a condition where it has no choice nevertheless to slow down. The constant entering and deleting of temporary files that occur when we surf the Web leave our registries with thousands of false indicators inside our operating system's registry.<br><br>Another normal cause of PC slow down is a corrupt registry. The registry is a rather important component of computers running on Windows platform. When this gets corrupted the PC will slowdown, or worse, not start at all. Fixing the registry is easy with the use of a program plus [http://bestregistrycleanerfix.com/regzooka regzooka].<br><br>If you think which there are issues with all the d3d9.dll file, then you must replace it with a fresh functioning file. This is performed by conducting a series of procedures plus we can commence by downloading "d3d9.zip" within the server. Next you must unzip the "d3d9.dll" file found on the hard drive of your computer. Proceed by locating "C:\Windows\System32" plus then acquiring the existing "d3d9.dll" on the PC. Once found, rename the file "d3d9.dll to d3d9BACKUP.dll" plus then copy-paste this fresh file to "C:\Windows\System32". After that, press "Start" followed by "Run" or look "Run" on Windows Vista & 7. As soon because a box shows up, kind "cmd". A black screen may then appear and you must type "regsvr32d3d9.dll" plus then click "Enter". This task might allow you to replace the aged file with all the fresh copy.<br><br>The System File Checker (SFC) may help in resolving error 1721 as it, by its nature, scans the program files for corruption plus replaces them with their authentic versions. This requires we to have the Windows Installation DVD ROM for continuing.<br><br>Before you buy a whole new system; it is actually time to get the old one cleaned up so you can begin getting more done online today! Visit our website under and access the most reputable registry cleaner software available. |
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| ==Origin of the constant==
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| The derivative of any constant function is zero. Once one has found one antiderivative <math>F(x)</math> for a function <math>f(x)</math>, adding or subtracting any constant ''C'' will give us another antiderivative, because <math>(F(x) + C)' = F\,'(x) + C\,' = F\,'(x)</math>. The constant is a way of expressing that every function has an infinite number of different antiderivatives.
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| For example, suppose one wants to find antiderivatives of <math>\cos(x)</math>. One such antiderivative is <math>\sin(x)</math>. Another one is <math>\sin(x)+1</math>. A third is <math>\sin(x)-\pi</math>. Each of these has derivative <math>\cos(x)</math>, so they are all antiderivatives of <math>\cos(x)</math>.
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| It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(''x''), we write:
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| :<math>\int \cos(x)\,dx = \sin(x) + C.</math>
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| Replacing ''C'' by a number will produce an antiderivative. By writing ''C'' instead of a number, however, a compact description of all the possible antiderivatives of cos(''x'') is obtained. ''C'' is called the '''constant of integration'''. It is easily determined that all of these functions are indeed antiderivatives of <math>\cos(x)</math>:
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| :<math>\begin{align}
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| \frac{d}{dx}[\sin(x) + C] &= \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[C] \\
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| &= \cos(x) + 0 \\
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| &= \cos(x)
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| \end{align}</math>
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| ==Necessity of the constant==
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| At first glance it may seem that the constant is unnecessary, since it can be set to zero. Furthermore, when evaluating [[definite integral]]s using the [[fundamental theorem of calculus]], the constant will always cancel with itself.
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| However, trying to set the constant equal to zero doesn't always make sense. For example, <math>2\sin(x)\cos(x)</math> can be integrated in at least three different ways:
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| :<math>\begin{align}
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| \int 2\sin(x)\cos(x)\,dx &=& \sin^2(x) + C &=& -\cos^2(x) + 1 + C &=& -\frac12\cos(2x) + C\\
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| \int 2\sin(x)\cos(x)\,dx &=& -\cos^2(x) + C &=& \sin^2(x) - 1 + C &=& -\frac12\cos(2x) + C\\
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| \int 2\sin(x)\cos(x)\,dx &=& -\frac12\cos(2x) + C &=& \sin^2(x) + C &=& -\cos^2(x) + C
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| \end{align}</math>
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| So setting ''C'' to zero can still leave a constant. This means that, for a given function, there is no "simplest antiderivative".
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| Another problem with setting ''C'' equal to zero is that sometimes we want to find an antiderivative that has a given value at a given point (as in an [[initial value problem]]). For example, to obtain the antiderivative of <math>\cos(x)</math> that has the value 100 at ''x'' = π, then only one value of ''C'' will work (in this case ''C'' = 100).
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| This restriction can be rephrased in the language of [[differential equations]]. Finding an indefinite integral of a function <math>f(x)</math> is the same as solving the differential equation <math>\frac{dy}{dx} = f(x)</math>. Any differential equation will have many solutions, and each constant represents the unique solution of a well-posed [[initial value problem]]. Imposing the condition that our antiderivative takes the value 100 at ''x'' = π is an initial condition. Each initial condition corresponds to one and only one value of ''C'', so without ''C'' it would be impossible to solve the problem.
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| There is another justification, coming from [[abstract algebra]]. The space of all (suitable) real-valued functions on the [[real number]]s is a [[vector space]], and the [[differential operator]] <math>\frac{d}{dx}</math> is a [[linear operator]]. The operator<math>\frac{d}{dx}</math> maps a function to zero if and only if that function is constant. Consequently, the [[kernel (algebra)|kernel]] of <math>\frac{d}{dx}</math> is the space of all constant functions. The process of indefinite integration amounts to finding a preimage of a given function. There is no canonical preimage for a given function, but the set of all such preimages forms a [[coset]]. Choosing a constant is the same as choosing an element of the coset. In this context, solving an [[initial value problem]] is interpreted as lying in the [[hyperplane]] given by the [[initial condition]]s.
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| ==Reason for a constant difference between antiderivatives ==
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| This result can be formally stated in this manner: Let <math>F:\mathbb{R}\rightarrow\mathbb{R}</math> and <math>G:\mathbb{R}\rightarrow\mathbb{R}</math> be two everywhere differentiable functions. Suppose that <math>F\,'(x) = G\,'(x)</math> for every real number ''x''. Then there exists a real number ''C'' such that <math>F(x) - G(x) = C</math> for every real number ''x''.
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| To prove this, notice that <math>[F(x) - G(x)]' = 0</math>. So ''F'' can be replaced by ''F''-''G'' and ''G'' by the constant function 0, making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant:
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| Choose a real number ''a'', and let <math>C = F(a)</math>. For any ''x'', the [[fundamental theorem of calculus]] says that
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| :<math>\begin{align}
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| \int_a^x 0\,dt &= F(x)-F(a)\\
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| &= F(x)-C,
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| \end{align}</math>
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| which implies that <math>F(x)=C</math>. So ''F'' is a constant function.
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| Two facts are crucial in this proof. First, the real line is [[connected space|connected]]. If the real line were not connected, we would not always be able to integrate from our fixed ''a'' to any given ''x''. For example, if we were to ask for functions defined on the union of intervals [0,1] and [2,3], and if ''a'' were 0, then it would not be possible to integrate from 0 to 3, because the function is not defined between 1 and 2. Here there will be ''two'' constants, one for each [[connected space|connected component]] of the [[function domain|domain]]. In general, by replacing constants with [[locally constant function]]s, we can extend this theorem to disconnected domains. For example, there are two constants of integration for <math>\textstyle\int dx/x</math> and infinitely many for <math>\textstyle\int \tan x\,dx,</math> so for example the general form for the integral of 1/''x'' is:<ref>"[http://golem.ph.utexas.edu/category/2012/03/reader_survey_logx_c.html Reader Survey: log|''x''| + ''C'']", Tom Leinster, ''The ''n''-category Café'', March 19, 2012</ref>
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| :<math>\int {1 \over x}\,dx = \begin{cases}\ln \left|x \right| + C^- & x < 0\\ | |
| \ln \left|x \right| + C^+ & x > 0
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| \end{cases}</math>
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| Second, ''F'' and ''G'' were assumed to be everywhere differentiable. If ''F'' and ''G'' are not differentiable at even one point, the theorem fails. As an example, let <math>F(x)</math> be the [[Heaviside step function]], which is zero for negative values of ''x'' and one for non-negative values of ''x'', and let <math>G(x)=0</math>. Then the derivative of ''F'' is zero where it is defined, and the derivative of ''G'' is always zero. Yet it's clear that ''F'' and ''G'' do not differ by a constant.
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| Even if it is assumed that ''F'' and ''G'' are everywhere continuous and [[almost everywhere]] differentiable the theorem still fails. As an example, take ''F'' to be the [[Cantor function]] and again let ''G'' = 0.
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| ==References==
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| {{reflist}}
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| [[Category:Integral calculus]]
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If your computer is running slow, you have probably gone from the different stages of rage plus frustration. Having such a excellent tool like a computer could seem like a curse along with a blessing simultaneously whenever this happens. It is fantastic when it is actually running rapidly and smooth, nevertheless then when it begins acting strange and slows way down, frustration sets inside. How will something because wonderful because a computer make a person so mad?
We all understand which the registry is the important component of the Windows running program as it stores all info about the Dll files, programs on the computer plus system settings. However, as days by, it really is unavoidable which we could encounter registry matter due to a huge amount of invalid, useless plus unwanted entries.
H/w connected error handling - when hardware causes BSOD installing newest fixes for the hardware and/ or motherboard will help. We could additionally add fresh hardware which is compatible with all the system.
There can be several factors why a computer can lose speed. Normal computer utilize, such as surfing the Internet may receive a running system in a condition where it has no choice nevertheless to slow down. The constant entering and deleting of temporary files that occur when we surf the Web leave our registries with thousands of false indicators inside our operating system's registry.
Another normal cause of PC slow down is a corrupt registry. The registry is a rather important component of computers running on Windows platform. When this gets corrupted the PC will slowdown, or worse, not start at all. Fixing the registry is easy with the use of a program plus regzooka.
If you think which there are issues with all the d3d9.dll file, then you must replace it with a fresh functioning file. This is performed by conducting a series of procedures plus we can commence by downloading "d3d9.zip" within the server. Next you must unzip the "d3d9.dll" file found on the hard drive of your computer. Proceed by locating "C:\Windows\System32" plus then acquiring the existing "d3d9.dll" on the PC. Once found, rename the file "d3d9.dll to d3d9BACKUP.dll" plus then copy-paste this fresh file to "C:\Windows\System32". After that, press "Start" followed by "Run" or look "Run" on Windows Vista & 7. As soon because a box shows up, kind "cmd". A black screen may then appear and you must type "regsvr32d3d9.dll" plus then click "Enter". This task might allow you to replace the aged file with all the fresh copy.
The System File Checker (SFC) may help in resolving error 1721 as it, by its nature, scans the program files for corruption plus replaces them with their authentic versions. This requires we to have the Windows Installation DVD ROM for continuing.
Before you buy a whole new system; it is actually time to get the old one cleaned up so you can begin getting more done online today! Visit our website under and access the most reputable registry cleaner software available.