What is the integral of (-x^{-3}+1)/(x^3) ?

The integral of (-x^{-3}+1)/(x^3) is 1/(5x^5)-1/(2x^2)+C

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integral of (-x^{-3}+1)/(x^3)

\int \frac{- x ^{- 3} + 1}{x ^{3}} d x

\frac{1}{5 x ^{5}} - \frac{1}{2 x ^{2}} + C

Solution steps

\int \frac{- x ^{- 3} + 1}{x ^{3}} d x

Simplify

\frac{- x ^{- 3} + 1}{x ^{3}} : \frac{- 1 + x ^{3}}{x ^{6}}

= \int \frac{- 1 + x ^{3}}{x ^{6}} d x

Expand

\frac{- 1 + x ^{3}}{x ^{6}} : - \frac{1}{x ^{6}} + \frac{1}{x ^{3}}

= \int - \frac{1}{x ^{6}} + \frac{1}{x ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{x ^{6}} d x + \int \frac{1}{x ^{3}} d x

\int \frac{1}{x ^{6}} d x = - \frac{1}{5 x ^{5}}

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

= - (- \frac{1}{5 x ^{5}}) - \frac{1}{2 x ^{2}}

Simplify

= \frac{1}{5 x ^{5}} - \frac{1}{2 x ^{2}}

Add a constant to the solution

= \frac{1}{5 x ^{5}} - \frac{1}{2 x ^{2}} + C

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What is the integral of (-x^{-3}+1)/(x^3) ?
The integral of (-x^{-3}+1)/(x^3) is 1/(5x^5)-1/(2x^2)+C