What is the integral of (x^7+x^5-4x^3)/(x^6) ?

The integral of (x^7+x^5-4x^3)/(x^6) is (x^2)/2+ln|x|+2/(x^2)+C

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integral of (x^7+x^5-4x^3)/(x^6)

\int \frac{x ^{7} + x ^{5} - 4 x ^{3}}{x ^{6}} d x

\frac{x ^{2}}{2} + ln ∣ x ∣ + \frac{2}{x ^{2}} + C

Solution steps

\int \frac{x ^{7} + x ^{5} - 4 x ^{3}}{x ^{6}} d x

Simplify

\frac{x ^{7} + x ^{5} - 4 x ^{3}}{x ^{6}} : \frac{x ^{4} + x ^{2} - 4}{x ^{3}}

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= \frac{x ^{2}}{2} + ln ∣ x ∣ - (- \frac{2}{x ^{2}})

Simplify

= \frac{x ^{2}}{2} + ln ∣ x ∣ + \frac{2}{x ^{2}}

Add a constant to the solution

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

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What is the integral of (x^7+x^5-4x^3)/(x^6) ?
The integral of (x^7+x^5-4x^3)/(x^6) is (x^2)/2+ln|x|+2/(x^2)+C