What is the integral of (2x-18)/(sqrt(x)+3) ?

The integral of (2x-18)/(sqrt(x)+3) is -6x+4/3 x^{3/2}-108+C

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integral of (2x-18)/(sqrt(x)+3)

\int \frac{2 x - 18}{x + 3} d x

- 6 x + \frac{4}{3} x^{\frac{3}{2}} - 108 + C

Solution steps

\int \frac{2 x - 18}{x + 3} d x

Expand

\frac{2 x - 18}{x + 3} : \frac{2 x}{x + 3} - \frac{18}{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{2 x}{x + 3} d x - \int \frac{18}{x + 3} d x

\int \frac{2 x}{x + 3} d x = 4 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3)

\int \frac{18}{x + 3} d x = 36 (x + 3 - 3 ln x + 3)

= 4 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3) - 36 (x + 3 - 3 ln x + 3)

Simplify

4 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3) - 36 (x + 3 - 3 ln x + 3) : - 6 x + \frac{4}{3} x^{\frac{3}{2}} - 108

= - 6 x + \frac{4}{3} x^{\frac{3}{2}} - 108

Add a constant to the solution

= - 6 x + \frac{4}{3} x^{\frac{3}{2}} - 108 + C

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What is the integral of (2x-18)/(sqrt(x)+3) ?
The integral of (2x-18)/(sqrt(x)+3) is -6x+4/3 x^{3/2}-108+C