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

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

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

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

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

Solution steps

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

\int \frac{x}{x + 2} d x = x + 2 - 2 ln ∣ x + 2 ∣

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

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

Add a constant to the solution

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

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