What is the integral of (x^{-3}+2x^5)/(x^4) ?

The integral of (x^{-3}+2x^5)/(x^4) is -1/(6x^6)+x^2+C

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integral of (x^{-3}+2x^5)/(x^4)

\int \frac{x ^{- 3} + 2 x ^{5}}{x ^{4}} d x

- \frac{1}{6 x ^{6}} + x^{2} + C

Solution steps

\int \frac{x ^{- 3} + 2 x ^{5}}{x ^{4}} d x

Expand

\frac{x ^{- 3} + 2 x ^{5}}{x ^{4}} : \frac{1}{x ^{7}} + 2 x

= \int \frac{1}{x ^{7}} + 2 x 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 ^{7}} d x + \int 2 x d x

\int \frac{1}{x ^{7}} d x = - \frac{1}{6 x ^{6}}

\int 2 x d x = x^{2}

= - \frac{1}{6 x ^{6}} + x^{2}

Add a constant to the solution

= - \frac{1}{6 x ^{6}} + x^{2} + C

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What is the integral of (x^{-3}+2x^5)/(x^4) ?
The integral of (x^{-3}+2x^5)/(x^4) is -1/(6x^6)+x^2+C