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Why is it important to add the constant of integration immediately when the integration is performed?
Wiki User
∙ 13y ago
When you do an integration, you are (implicitly or explicitly) recognizing that what you are integrating is a "rate of change". Your integration over a particular interval provides you with the answer to the question "what is the total change over this interval?". To get the total value of this quantity you must add the initial amount or value. That is represented by the constant of integration.
When you integrate between specific limits and you are asking the question "how much is the total change" the initial value is not needed, and in fact does not appear when you insert the initial and final values of the variable over which you are integrating.
So you must distinguish between finding the total change, or finding the final value.
Re-reading this, I could have been a bit clearer. I'll give an example. Suppose something is accelerating at a constant acceleration designated by "a". Between the times t1 and t2 the velocity changes by a(t2-t1) which you get by integrating "a" and applying the limits t2 and t1. But the change in velocity is not the same as the velocity itself, which is equal to the initial velocity, "vo", plus the change in velocity a(t2-t1). This shows that the integral between limits just gives the accumulated change. but if you want the final VALUE, you have to add on the initial value. You might see a statement like "the integral of a with respect to time, when a is constant is vo + at ". You can check this by differentiating with respect to t, and you find the constant vo disappears.
In summary, the integral evaluated by simply applying the limits gives the accumulated change, but to get the final value you have to add on the
pre-existing value, and in this context the pre-existing value also carries the name of "constant of integration".
Wiki User
∙ 13y ago
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How do you integrate a constant in calculus For ex S12dx?
constants can be brought outside the integration symbol as a constant multiplier. for example: S12dx is the same as 12Sdx which is the same as 12x+C however: S(x+3)dx is Sxdx+S3dx not 3Sxdx
What is the integral of the derivative with respect to x of a function of x with respect to x?
∫ d/dx f(x) dx = f(x) + C C is the constant of integration.
What is the integral of 1 divided by the cosine of x with respect to x?
∫ 1/cos(x) dx = ln(sec(x) + tan(x)) + C C is the constant of integration.
Does f-gx equals g-fx?
Yes, the integral of gx dx is g integral x dx. In this case, g is unrelated to x, so it can be treated as constant and pulled outside of the integration.
Ten uses of integration in calculus?
Integration can be used whenever you have to multiply two numbers, one of which varies - for example, to calculate an area, you calculate height times width, but the height may vary in certain geometric figures.Integration can also be used to calculate:Work = force times distance (force may not be constant). Center of mass - you need to take the average of many pieces of massMoment of inertiaArea of a surfaceVolumesAnd many others more.Integration can be used whenever you have to multiply two numbers, one of which varies - for example, to calculate an area, you calculate height times width, but the height may vary in certain geometric figures. Integration can also be used to calculate:Work = force times distance (force may not be constant).Center of mass - you need to take the average of many pieces of massMoment of inertiaArea of a surfaceVolumesAnd many others more.Integration can be used whenever you have to multiply two numbers, one of which varies - for example, to calculate an area, you calculate height times width, but the height may vary in certain geometric figures. Integration can also be used to calculate:Work = force times distance (force may not be constant).Center of mass - you need to take the average of many pieces of massMoment of inertiaArea of a surfaceVolumesAnd many others more.Integration can be used whenever you have to multiply two numbers, one of which varies - for example, to calculate an area, you calculate height times width, but the height may vary in certain geometric figures. Integration can also be used to calculate:Work = force times distance (force may not be constant).Center of mass - you need to take the average of many pieces of massMoment of inertiaArea of a surfaceVolumesAnd many others more.
Why is it important to introduce constant of integration immediately when the integration is performed?
I'm assuming you are asking why you cannot work through your simplification and only put a constant on the last line. The simplest answer is that mathematicians are picky people, and when working through a problem EVERY line must make absolute mathematical sense. Leaving the constant off until the last line makes every line between the point where the integration occurs and the last line false. (Unless you are lucky and the constant of integration is 0, however this still needs to be proven)
What is the integration of sinehx?
It is cosh(x) + c where c is a constant of integration.
What is integration of 34?
Assuming integration is with respect to a variable, x, the answer is 34x + c where c is the constant of integration.
Why do you need initial condition to solve differential equation?
The solution to a differential equation requires integration. With any integration, there is a constant of integration. This constant can only be found by using additional conditions: initial or boundary.
How do you find constant in math?
The answer depends on what the constant is: the y-intercept in a linear graph, constant of proportionality, constant of integration, physical [universal] constant.
What is the integral of ln2?
ln2 is a constant so x*ln2 + c, where c is the constant of integration.
What is tahe integration of y?
1/2 y2 + any constant
What is the integral of a constant to the power of x with respect to x?
∫ ax dx = ax/ln(a) + C C is the constant of integration.
What is the antiderivative of 1 divided by 18x5?
-1/72x4 + c where c is the constant of integration.
Why does an answer to an integration problem involve a Constant of Integration?
The indefinite integral is the anti-derivative - so the question is, "What function has this given function as a derivative". And if you add a constant to a function, the derivative of the function doesn't change. Thus, for example, if the derivative is y' = 2x, the original function might be y = x squared. However, any function of the form y = x squared + c (for any constant c) also has the SAME derivative (2x in this case). Therefore, to completely specify all possible solutions, this constant should be added.
Why a constant is written after integrating?
Integration is the opposite of differentiation (taking the derivative). The derivative of a constant is zero. Integration is also called antidifferentiation since integration and differentiation are opposites of each other. The derivative of x^2 is 2x. The antiderivative (integral) of 2x is x^2. However, the derivative of x^2 + 7 is also 2x. Therefore, the antiderivative of 2x is x^2 + C, in general, where the constant C has to be determined from the context of the problem. In the above case, the constant happens to be C=7. We use integration to solve first order differential equations. When solving first order differential equations, like in "word problems", you must determine the integration constant using the initial conditions (ie the conditions we know to be true at t=0 - we usually know what these are), or the boundary conditions (ie the conditions we know to be true at x=0 and y=0).
What is the integral of e to the power of x with respect to x?
∫ ex dx = ex + CC is the constant of integration.
