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What is the antiderivative of a constant?

Answer 1

Christopher Allers

I find it simpler to think of this looking at the derivative first.

I mean: given a first degree variable, what would be the outcome of differentiation that would yield a constant?

For example, if your differentiation resulted in #f'(x)=5#, it's evident that the antiderivative is #F(x)=5x#

So, the antiderivative of a constant is it times the variable in question (be it #x#, #y#, etc.)

We could express it mathematically as follows:

#intcdx <=> cx#

Note that #c# is mutiplying #1# in the integral:

#intcolor(green)(1)*cdx <=> cx#

This indicates that a first-degree variable is being distinguished:

#f(x)=x^color(green)(1)#, then #f'(x)=color(green)1*x^(1-1)=1*x^0=color(green)(1)#

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Answer 2

Christopher Agresta

The antiderivative of a constant is simply the variable multiplied by that constant, plus an arbitrary constant of integration. In mathematical notation, if ( C ) is a constant, then the antiderivative of ( C ) with respect to a variable ( x ) is ( Cx + K ), where ( K ) represents the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.