What is the integral of (4x-16)/(sqrt(x)+2) ?

The integral of (4x-16)/(sqrt(x)+2) is -24x-32sqrt(x)+8/3 (sqrt(x)+2)^3+32+C

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integral of (4x-16)/(sqrt(x)+2)

\int \frac{4 x - 16}{x + 2} d x

- 24 x - 32 x + \frac{8}{3} (x + 2)^{3} + 32 + C

Solution steps

\int \frac{4 x - 16}{x + 2} d x

Expand

\frac{4 x - 16}{x + 2} : \frac{4 x}{x + 2} - \frac{16}{x + 2}

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{4 x}{x + 2} d x - \int \frac{16}{x + 2} d x

\int \frac{4 x}{x + 2} d x = 8 (- 3 x + \frac{1}{3} (x + 2)^{3} - 8 ln x + 2 + 12)

\int \frac{16}{x + 2} d x = 32 (x + 2 - 2 ln x + 2)

= 8 (- 3 x + \frac{1}{3} (x + 2)^{3} - 8 ln x + 2 + 12) - 32 (x + 2 - 2 ln x + 2)

Simplify

8 (- 3 x + \frac{1}{3} (x + 2)^{3} - 8 ln x + 2 + 12) - 32 (x + 2 - 2 ln x + 2) : - 24 x - 32 x + \frac{8}{3} (x + 2)^{3} + 32

= - 24 x - 32 x + \frac{8}{3} (x + 2)^{3} + 32

Add a constant to the solution

= - 24 x - 32 x + \frac{8}{3} (x + 2)^{3} + 32 + C

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What is the integral of (4x-16)/(sqrt(x)+2) ?
The integral of (4x-16)/(sqrt(x)+2) is -24x-32sqrt(x)+8/3 (sqrt(x)+2)^3+32+C