What is the integral of (x+2)(3x-1/x) ?

The integral of (x+2)(3x-1/x) is x^3-x+3x^2-2ln|x|+C

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integral of (x+2)(3x-1/x)

\int (x + 2) (3 x - \frac{1}{x}) d x

x^{3} - x + 3 x^{2} - 2 ln ∣ x ∣ + C

Solution steps

\int (x + 2) (3 x - \frac{1}{x}) d x

Simplify

(x + 2) (3 x - \frac{1}{x}) : 3 x^{2} - 1 + 6 x - \frac{2}{x}

= \int 3 x^{2} - 1 + 6 x - \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 1 d x + \int 6 x d x - \int \frac{2}{x} d x

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

\int 1 d x = x

\int 6 x d x = 3 x^{2}

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= x^{3} - x + 3 x^{2} - 2 ln ∣ x ∣

Add a constant to the solution

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

What is the integral of (x+2)(3x-1/x) ?
The integral of (x+2)(3x-1/x) is x^3-x+3x^2-2ln|x|+C