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The antiderivative of 1/x is ln(x) + C. That is, to the natural (base-e) logarithm, you can add any constant, and still have an antiderivative. For example, ln(x) + 5. These are the only antiderivatives; there are no different functions that have the same derivatives. This is valid, in general, for all antiderivatives: if you have one antiderivative of a function, all other antiderivatives are obtained by adding a constant.
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What is the antiderivative of pi?
The anti-derivative of any constant c, is just c*x. Thus, the antiderivative of pi is pi*x. We can verify this by taking the derivative of pi*x, which gives us pi.
What does the constant rate mean in math?
It means that the first derivative is a constant. The derivative may be with regard to time or any other variable.
What is the derivative of 40?
The derivative of 40 is zero. The derivative of any constant is zero.
What is the derivative of 8?
Zero. In general, the derivative of any constant is zero.
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.
