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

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

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

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

x^{3} - 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= x^{3} - 2 ln ∣ x ∣

Add a constant to the solution

= x^{3} - 2 ln ∣ x ∣ + C

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