What is the integral of (x^3-4x^2+8x+3)/x ?

The integral of (x^3-4x^2+8x+3)/x is (x^3)/3-2x^2+8x+3ln|x|+C

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integral of (x^3-4x^2+8x+3)/x

\int \frac{x ^{3} - 4 x ^{2} + 8 x + 3}{x} d x

\frac{x ^{3}}{3} - 2 x^{2} + 8 x + 3 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{3} - 4 x ^{2} + 8 x + 3}{x} d x

Expand

\frac{x ^{3} - 4 x ^{2} + 8 x + 3}{x} : x^{2} - 4 x + 8 + \frac{3}{x}

= \int x^{2} - 4 x + 8 + \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^{2} d x - \int 4 x d x + \int 8 d x + \int \frac{3}{x} d x

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

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

\int 8 d x = 8 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= \frac{x ^{3}}{3} - 2 x^{2} + 8 x + 3 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 x^{2} + 8 x + 3 ln ∣ x ∣ + C

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What is the integral of (x^3-4x^2+8x+3)/x ?
The integral of (x^3-4x^2+8x+3)/x is (x^3)/3-2x^2+8x+3ln|x|+C