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

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

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

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

2 x - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

4 - x = (2 + x) (2 - x)

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

Cancel the common factor:

2 + x

= \int 2 - x 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 2 d x - \int x d x

\int 2 d x = 2 x

\int x d x = \frac{2}{3} x^{\frac{3}{2}}

= 2 x - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

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

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