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

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

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

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

\frac{5 x ^{2}}{2} - 4 x + 3 ln ∣ x ∣ + C

Solution steps

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

Simplify

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

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

Expand

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

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

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

\int 4 d x = 4 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= \frac{5 x ^{2}}{2} - 4 x + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{5 x ^{2}}{2} - 4 x + 3 ln ∣ x ∣ + C

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