What is the integral of ((5-4x^2+2x^3-x^7))/(x^3) ?

The integral of ((5-4x^2+2x^3-x^7))/(x^3) is -5/(2x^2)-4ln|x|+2x-(x^5)/5+C

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integral of ((5-4x^2+2x^3-x^7))/(x^3)

\int \frac{( 5 - 4 x ^{2} + 2 x ^{3} - x ^{7} )}{x ^{3}} d x

- \frac{5}{2 x ^{2}} - 4 ln ∣ x ∣ + 2 x - \frac{x ^{5}}{5} + C

Solution steps

\int \frac{5 - 4 x ^{2} + 2 x ^{3} - x ^{7}}{x ^{3}} d x

Expand

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

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

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 2 d x = 2 x

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

= - \frac{5}{2 x ^{2}} - 4 ln ∣ x ∣ + 2 x - \frac{x ^{5}}{5}

Add a constant to the solution

= - \frac{5}{2 x ^{2}} - 4 ln ∣ x ∣ + 2 x - \frac{x ^{5}}{5} + C

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What is the integral of ((5-4x^2+2x^3-x^7))/(x^3) ?
The integral of ((5-4x^2+2x^3-x^7))/(x^3) is -5/(2x^2)-4ln|x|+2x-(x^5)/5+C