What is the integral of (x^7-5x)/(x^5) ?

The integral of (x^7-5x)/(x^5) is (x^3)/3+5/(3x^3)+C

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integral of (x^7-5x)/(x^5)

\int \frac{x ^{7} - 5 x}{x ^{5}} d x

\frac{x ^{3}}{3} + \frac{5}{3 x ^{3}} + C

Solution steps

\int \frac{x ^{7} - 5 x}{x ^{5}} d x

Expand

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

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

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

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

= \frac{x ^{3}}{3} - (- \frac{5}{3 x ^{3}})

Simplify

= \frac{x ^{3}}{3} + \frac{5}{3 x ^{3}}

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{5}{3 x ^{3}} + C

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