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

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

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

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

\frac{x ^{2}}{2} + 3 ln ∣ x ∣ + C

Solution steps

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

Simplify

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

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

Expand

\frac{x ^{2} + 3}{x} : x + \frac{3}{x}

= \int x + \frac{3}{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 x d x + \int \frac{3}{x} d x

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= \frac{x ^{2}}{2} + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 3 ln ∣ x ∣ + C

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