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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

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

Add a constant to the solution

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

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