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

The integral of (x-4)/(sqrt(x+2)) is 2/3 (x+2)^{3/2}-12sqrt(x+2)+C

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

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

\frac{2}{3} (x + 2)^{\frac{3}{2}} - 12 x + 2 + C

Solution steps

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

Expand

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

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

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

\int \frac{4}{x + 2} d x = 8 x + 2

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

Simplify

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

= \frac{2}{3} (x + 2)^{\frac{3}{2}} - 12 x + 2

Add a constant to the solution

= \frac{2}{3} (x + 2)^{\frac{3}{2}} - 12 x + 2 + C

What is the integral of (x-4)/(sqrt(x+2)) ?
The integral of (x-4)/(sqrt(x+2)) is 2/3 (x+2)^{3/2}-12sqrt(x+2)+C