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In order for your integration to be complete it has to represent the fact that it has infinite solutions, or else there's a possibility to have fallacies in the proofs you write. In other words, by neglecting the constant in your answer, it may be possible to extrapolate erroneous proofs from your incomplete answer, like 1 + 1 = 3, for example.
A little more practically, the constants of integration are great for bookkeeping when dealing with multiple integrals.
For example, here's the result from a simple triple integral without adding in the constants of integration:
∫∫∫ (xyz) dxdydz = ∫∫ (yzx2/2) dydz = ∫ (x2y2z/4) dz = x2y2z2/8
Whereas, with the constants added in you get this result:
∫∫∫ (xyz) dxdydz = ∫∫ (yzx2/2 + C) dydz = ∫ (x2y2z/4 + Cy + D) dz =
x2y2z2/8 + Cyz + Dz + E, where C, D, and E are the constants of integration.
This result has a term with both yz and z in it that we had initially missed, which could have had crucial applications to whatever this function is describing.
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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.
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.
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 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.
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.
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
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 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 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.
