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How do you find the antiderivative of #f(x)=x^5#?

Answer 1

Have a look:

You are looking for a function that once derived will give you #x^5#; a good candidate would be #x^6# in fact if you derive it you get: #6x^5#. it is not really equal but we can use a little trick and say (to get rid of the #6#) that our antiderivative will be: #F(x)=1/6x^6+c# the #c# is a constant that we add to be sure to cover all the possibilities and anyway, during the derivation, it will desappear (our antiderivative could be, for example, #F(x)=1/6x^6+7# or #F(x)=1/6x^6+345# or other number so that when you derive it you will still get #x^5#!!!).

As a general rule the antiderivative of #x^n# can be found as:

#x^(n+1)/(n+1)+c#

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Answer 2

William Abdalla

To find the antiderivative of ( f(x) = x^5 ), integrate ( x^5 ) with respect to ( x ). The antiderivative is ( F(x) = \frac{1}{6}x^6 + C ), where ( C ) is 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.