Question

What is the antiderivative of `1 / (x^5)`?

Answer 1

`"antiderivative "-> -1/(4x^4)+C`

Explanation:

Apply the revers of differentiation

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Example: antiderivative of `x^a-> ( x^(a+1))/(a+1)+C`

This is because `d/(dx) (1/(a+1) x^(a+1)+C) = (a+1)/(a+1) x^(a+1-1) = x^a`

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`color(white)(.)`

Write as `x^(-5)`

`=>"antiderivative" ->1/(-5+1)x^(-5+1) +C" " =" "1/(-4)x^-4+C`

`color(blue)(= -1/(4x^4)+C) color(red)(" "larr" Do not forget the constant."`

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Check:

`d/(dx) (-1/(4x^4)+C) -> d/(dx) (-(x^(-4))/4+C)`

`= (-4)(-(x^(-5))/4)" " =" "x^(-5)" "=" "1/x^5`

Which is where we started from so ok!

(Think of antiderivative as integration)