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

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

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

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

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

Solution steps

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

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

Add a constant to the solution

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

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