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

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

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

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

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

Solution steps

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

Expand

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

= \int 4 x^{2} + 2 x - 1 + \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 4 x^{2} d x + \int 2 x d x - \int 1 d x + \int \frac{3}{x} d x

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

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

Add a constant to the solution

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

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