What is the integral of (1/(2x)+sqrt(x)+1/(x^2)) ?

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

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integral of (1/(2x)+sqrt(x)+1/(x^2))

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

\frac{1}{2} ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{x} + C

Solution steps

\int \frac{1}{2 x} + x + \frac{1}{x ^{2}} 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{1}{2 x} d x + \int x d x + \int \frac{1}{x ^{2}} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

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

= \frac{1}{2} ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{x}

Add a constant to the solution

= \frac{1}{2} ln ∣ x ∣ + \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{x} + C

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