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I would think so, but I am sitting here thinking and I can not think of any function that can not be integrated.
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Whether or not a function can be integrated depends, in part, on the measure that is used on the space. This is getting into seriously heavy mathematics. and the following is only a flavour of what I remember from 35 years ago!
Functions with asymptotic values in their domain cannot be integrated over that domain. For example, 1/x is defined everywhere except at x = 0 and so it cannot be integrated over any interval that contains x = 0. Otherwise it is ln(|x|) + c.
Another example is that of a function which has infinitely many discontinuities. For example:
f(x) = 1 if x is rational and
f(x) = 0 if x is irrational
over any arbitrary interval, is a perfectly well defined, indicator function. But under some measures, it cannot be integrated.
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What is the second derivative of a function's indefinite integral?
well, the second derivative is the derivative of the first derivative. so, the 2nd derivative of a function's indefinite integral is the derivative of the derivative of the function's indefinite integral. the derivative of a function's indefinite integral is the function, so the 2nd derivative of a function's indefinite integral is the derivative of the function.
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 null derivative?
A null derivative occurs when an increasing function does not have a derivative. This is most commonly seen in the question mark function.
What is the geometrical meaning for second derivative?
The Geometrical meaning of the second derivative is the curvature of the function. If the function has zero second derivative it is straight or flat.
What is the function if the first derivative of a function is a constant?
A linear function, for example y(x) = ax + b has the first derivative a.
