What is the integral of (x^{-2}-4)/(x^3) ?

The integral of (x^{-2}-4)/(x^3) is -1/(4x^4)+2/(x^2)+C

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integral of (x^{-2}-4)/(x^3)

\int \frac{x ^{- 2} - 4}{x ^{3}} d x

- \frac{1}{4 x ^{4}} + \frac{2}{x ^{2}} + C

Solution steps

\int \frac{x ^{- 2} - 4}{x ^{3}} d x

Expand

\frac{x ^{- 2} - 4}{x ^{3}} : \frac{1}{x ^{5}} - \frac{4}{x ^{3}}

= \int \frac{1}{x ^{5}} - \frac{4}{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 ^{5}} d x - \int \frac{4}{x ^{3}} d x

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

\int \frac{4}{x ^{3}} d x = - \frac{2}{x ^{2}}

= - \frac{1}{4 x ^{4}} - (- \frac{2}{x ^{2}})

Simplify

= - \frac{1}{4 x ^{4}} + \frac{2}{x ^{2}}

Add a constant to the solution

= - \frac{1}{4 x ^{4}} + \frac{2}{x ^{2}} + C

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What is the integral of (x^{-2}-4)/(x^3) ?
The integral of (x^{-2}-4)/(x^3) is -1/(4x^4)+2/(x^2)+C