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

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

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

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

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

Solution steps

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

Expand

\frac{r + 3}{r + 3} : \frac{r}{r + 3} + \frac{3}{r + 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{r}{r + 3} d r + \int \frac{3}{r + 3} d r

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

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

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

Add a constant to the solution

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

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