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

Jul 21, 2016

$\text{antiderivative } \to - \frac{1}{4 {x}^{4}} + C$

#### Explanation:

Apply the revers of differentiation

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Example: antiderivative of ${x}^{a} \to \frac{{x}^{a + 1}}{a + 1} + C$

This is because $\frac{d}{\mathrm{dx}} \left(\frac{1}{a + 1} {x}^{a + 1} + C\right) = \frac{a + 1}{a + 1} {x}^{a + 1 - 1} = {x}^{a}$

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$\textcolor{w h i t e}{.}$

Write as ${x}^{- 5}$

$\implies \text{antiderivative" ->1/(-5+1)x^(-5+1) +C" " =" } \frac{1}{- 4} {x}^{-} 4 + C$

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

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Check:

$\frac{d}{\mathrm{dx}} \left(- \frac{1}{4 {x}^{4}} + C\right) \to \frac{d}{\mathrm{dx}} \left(- \frac{{x}^{- 4}}{4} + C\right)$

$= \left(- 4\right) \left(- \frac{{x}^{- 5}}{4}\right) \text{ " =" "x^(-5)" "=" } \frac{1}{x} ^ 5$

Which is where we started from so ok!

(Think of antiderivative as integration)