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

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

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

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

23 (- arctan (\frac{x}{3}) + \frac{x}{3}) + 3 ln ∣ x + 3 ∣ + C

Solution steps

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

Expand

\frac{x + 3}{x + 3} : \frac{x}{x + 3} + \frac{3}{x + 3}

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

\int \frac{x}{x + 3} d x = 23 (- arctan (\frac{x}{3}) + \frac{x}{3})

\int \frac{3}{x + 3} d x = 3 ln ∣ x + 3 ∣

= 23 (- arctan (\frac{x}{3}) + \frac{x}{3}) + 3 ln ∣ x + 3 ∣

Add a constant to the solution

= 23 (- arctan (\frac{x}{3}) + \frac{x}{3}) + 3 ln ∣ x + 3 ∣ + C

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