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

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

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

\int \frac{( 4 - x ) ^{2}}{x} d x

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

Solution steps

\int \frac{( 4 - x ) ^{2}}{x} d x

Expand

\frac{( 4 - x ) ^{2}}{x} : \frac{16}{x} - 8 + x

= \int \frac{16}{x} - 8 + 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 \frac{16}{x} d x - \int 8 d x + \int x d x

\int \frac{16}{x} d x = 32 x

\int 8 d x = 8 x

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

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

Add a constant to the solution

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

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